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This slogan seems to have originated around 1991 with Rolf Landauer. It’s ricocheted around quantum information for the entire time I’ve been in the field, incanted in funding agency reports and popular articles and at the beginnings and ends of talks.

But what the hell does it mean?

There are many things it’s taken to mean, in my experience, that don’t make a lot of sense when you think about them—or else they’re vacuously true, or purely a matter of perspective, or not faithful readings of the slogan’s words.

For example, some people seem to use the slogan to mean something more like its converse: “physics is informational.” That is, the laws of physics are ultimately not about mass or energy or pressure, but about bits and computations on them. As I’ve often said, my problem with that view is less its audacity than its timidity! It’s like, what would the universe have to do in order not to be informational in this sense? “Information” is just a name we give to whatever picks out one element from a set of possibilities, with the “amount” of information given by the log of the set’s cardinality (and with suitable generalizations to infinite sets, nonuniform probability distributions, yadda yadda). So, as long as the laws of physics take the form of telling us that some observations or configurations of the world are possible and others are not, or of giving us probabilities for each configuration, no duh they’re about information!

Other people use “information is physical” to pour scorn on the idea that “information” could mean anything without some actual physical instantiation of the abstract 0’s and 1’s, such as voltage differences in a loop of wire. Here I certainly agree with the tautology that in order to exist physically—that is, be embodied in the physical world—a piece of information (like a song, video, or computer program) does need to be embodied in the physical world. But my inner Platonist slumps in his armchair when people go on to assert that, for example, it’s meaningless to discuss the first prime number larger than 1010^125, because according to post-1998 cosmology, one couldn’t fit its digits inside the observable universe.

If the cosmologists revise their models next week, will this prime suddenly burst into existence, with all the mathematical properties that one could’ve predicted for it on general grounds—only to fade back into the netherworld if the cosmologists revise their models again? Why would anyone want to use language in such a tortured way?

Yes, brains, computers, yellow books, and so on that encode mathematical knowledge comprise only a tiny sliver of the physical world. But it’s equally true that the physical world we observe comprises only a tiny sliver of mathematical possibility-space.

Still other people use “information is physical” simply to express their enthusiasm for the modern merger of physical and information sciences, as exemplified by quantum computing. Far be it from me to temper that enthusiasm: rock on, dudes!

Yet others use “information is physical” to mean that the rules governing information processing and transmission in the physical world aren’t knowable a priori, but can only be learned from physics. This is clearest in the case of quantum information, which has its own internal logic that generalizes the logic of classical information. But in some sense, we didn’t need quantum mechanics to tell us this! Of course the laws of physics have ultimate jurisdiction over whatever occurs in the physical world, information processing included.

My biggest beef, with all these unpackings of the “information is physical” slogan, is that none of them really engage with any of the deep truths that we’ve learned about physics. That is, we could’ve had more-or-less the same debates about any of them, even in a hypothetical world where the laws of physics were completely different.

So then what should we mean by “information is physical”? In the rest of this post, I’d like to propose an answer to that question.

We get closer to the meat of the slogan if we consider some actual physical phenomena, say in quantum mechanics. The double-slit experiment will do fine.

Recall: you shoot photons, one by one, at a screen with two slits, then examine the probability distribution over where the photons end up on a second screen. You ask: does that distribution contain alternating “light” and “dark” regions, the signature of interference between positive and negative amplitudes? And the answer, predicted by the math and confirmed by experiment, is: yes, but only if the information about which slit the photon went through failed to get recorded anywhere else in the universe, other than the photon location itself.

Here a skeptic interjects: but that has to be wrong! The criterion for where a physical particle lands on a physical screen can’t possibly depend on anything as airy as whether “information” got “recorded” or not. For what counts as “information,” anyway? As an extreme example: what if God, unbeknownst to us mortals, took divine note of which slit the photon went through? Would that destroy the interference pattern? If so, then every time we do the experiment, are we collecting data about the existence or nonexistence of an all-knowing God?

It seems to me that the answer is: insofar as the mind of God can be modeled as a tensor factor in Hilbert space, yes, we are. And crucially, if quantum mechanics is universally true, then the mind of God would have to be such a tensor factor, in order for its state to play any role in the prediction of observed phenomena.

To say this another way: it’s obvious and unexceptionable that, by observing a physical system, you can often learn something about what information must be in it. For example, you need never have heard of DNA to deduce that chickens must somehow contain information about making more chickens. What’s much more surprising is that, in quantum mechanics, you can often deduce things about what information can’t be present, anywhere in the physical world—because if such information existed, even a billion light-years away, it would necessarily have a physical effect that you don’t see.

Another famous example here concerns identical particles. You may have heard the slogan that “if you’ve seen one electron, you’ve seen them all”: that is, apart from position, momentum, and spin, every two electrons have exactly the same mass, same charge, same every other property, including even any properties yet to be discovered. Again the skeptic interjects: but that has to be wrong. Logically, you could only ever confirm that two electrons were different, by observing a difference in their behavior. Even if the electrons had behaved identically for a billion years, you couldn’t rule out the possibility that they were actually different, for example because of tiny nametags (“Hi, I’m Emily the Electron!” “Hi, I’m Ernie!”) that had no effect on any experiment you’d thought to perform, but were visible to God.

You can probably guess where this is going. Quantum mechanics says that, no, you can verify that two particles are perfectly identical by doing an experiment where you swap them and see what happens. If the particles are identical in all respects, then you’ll see quantum interference between the swapped and un-swapped states. If they aren’t, you won’t. The kind of interference you’ll see is different for fermions (like electrons) than for bosons (like photons), but the basic principle is the same in both cases. Once again, quantum mechanics lets you verify that a specific type of information—in this case, information that distinguishes one particle from another—was not present anywhere in the physical world, because if it were, it would’ve destroyed an interference effect that you in fact saw.

This, I think, already provides a meatier sense in which “information is physical” than any of the senses discussed previously.

But we haven’t gotten to the filet mignon yet. The late, great Jacob Bekenstein will forever be associated with the discovery that information, wherever and whenever it occurs in the physical world, takes up a minimum amount of space. The most precise form of this statement, called the covariant entropy bound, was worked out in detail by Raphael Bousso. Here I’ll be discussing a looser version of the bound, which holds in “non-pathological” cases, and which states that a bounded physical system can store at most A/(4 ln 2) bits of information, where A is the area in Planck units of any surface that encloses the system—so, about 1069 bits per square meter. (Actually it’s 1069qubits per square meter, but because of Holevo’s theorem, an upper bound on the number of qubits is also an upper bound on the number of classical bits that can be reliably stored in a system and then retrieved later.)

You might have heard of the famous way Nature enforces this bound. Namely, if you tried to create a hard drive that stored more than 1069 bits per square meter of surface area, the hard drive would necessarily collapse to a black hole. And from that point on, the information storage capacity would scale “only” with the area of the black hole’s event horizon—a black hole itself being the densest possible hard drive allowed by physics.

Let’s hear once more from our skeptic. “Nonsense! Matter can take up space. Energy can take up space. But information? Bah! That’s just a category mistake. For a proof, suppose God took one of your black holes, with a 1-square-meter event horizon, which already had its supposed maximum of ~1069 bits of information. And suppose She then created a bunch of new fundamental fields, which didn’t interact with gravity, electromagnetism, or any of the other fields that we know from observation, but which had the effect of encoding 10300 new bits in the region of the black hole. Presto! An unlimited amount of additional information, exactly where Bekenstein said it couldn’t exist.”

We’d like to pinpoint what’s wrong with the skeptic’s argument—and do so in a self-contained, non-question-begging way, a way that doesn’t pull any rabbits out of hats, other than the general principles of relativity and quantum mechanics. I was confused myself about how to do this, until a month ago, when Daniel Harlow helped set me straight (any remaining howlers in my exposition are 100% mine, not his).

I believe the logic goes like this:

Relativity—even just Galilean relativity—demands that, in flat space, the laws of physics must have the same form for all inertial observers (i.e., all observers who move through space at constant speed).

Anything in the physical world that varies in space—say, a field that encodes different bits of information at different locations—also varies in time, from the perspective of an observer who moves through the field at a constant speed.

Combining 1 and 2, we conclude that anything that can vary in space can also vary in time. Or to say it better, there’s only one kind of varying: varying in spacetime.

More strongly, special relativity tells us that there’s a specific numerical conversion factor between units of space and units of time: namely the speed of light, c. Loosely speaking, this means that if we know the rate at which a field varies across space, we can also calculate the rate at which it varies across time, and vice versa.

Anything that varies across time carries energy. Why? Because this is essentially the definition of energy in quantum mechanics! Up to a constant multiple (namely, Planck’s constant), energy is the expected speed of rotation of the global phase of the wavefunction, when you apply your Hamiltonian. If the global phase rotates at the slowest possible speed, then we take the energy to be zero, and say you’re in a vacuum state. If it rotates at the next highest speed, we say you’re in a first excited state, and so on. Indeed, assuming a time-independent Hamiltonian, the evolution of any quantum system can be fully described by simply decomposing the wavefunction into a superposition of energy eigenstates, then tracking of the phase of each eigenstate’s amplitude as it loops around and around the unit circle. No energy means no looping around means nothing ever changes.

Combining 3 and 5, any field that varies across space carries energy.

More strongly, combining 4 and 5, if we know how quickly a field varies across space, we can lower-bound how much energy it has to contain.

In general relativity, anything that carries energy couples to the gravitational field. This means that anything that carries energy necessarily has an observable effect: if nothing else, its effect on the warping of spacetime. (This is dramatically illustrated by dark matter, which is currently observable via its spacetime warping effect and nothing else.)

Combining 6 and 8, any field that varies across space couples to the gravitational field.

More strongly, combining 7 and 8, if we know how quickly a field varies across space, then we can lower-bound by how much it has to warp spacetime. This is so because of another famous (and distinctive) feature of gravity: namely, the fact that it’s universally attractive, so all the warping contributions add up.

But in GR, spacetime can only be warped by so much before we create a black hole: this is the famous Schwarzschild bound.

Combining 10 and 11, the information contained in a physical field can only vary so quickly across space, before it causes spacetime to collapse to a black hole.

Summarizing where we’ve gotten, we could say: any information that’s spatially localized at all, can only be localized so precisely. In our world, the more densely you try to pack 1’s and 0’s, the more energy you need, therefore the more you warp spacetime, until all you’ve gotten for your trouble is a black hole. Furthermore, if we rewrote the above conceptual argument in math—keeping track of all the G’s, c’s, h’s, and so on—we could derive a quantitative bound on how much information there can be in a bounded region of space. And if we were careful enough, that bound would be precisely the holographic entropy bound, which says that the number of (qu)bits is at most A/(4 ln 2), where A is the area of a bounding surface in Planck units.

Let’s pause to point out some interesting features of this argument.

Firstly, we pretty much needed the whole kitchen sink of basic physical principles: special relativity (both the equivalence of inertial frames and the finiteness of the speed of light), quantum mechanics (in the form of the universal relation between energy and frequency), and finally general relativity and gravity. All three of the fundamental constants G, c, and h made appearances, which is why all three show up in the detailed statement of the holographic bound.

But secondly, gravity only appeared from step 8 onwards. Up till then, everything could be said solely in the language of quantum field theory: that is, quantum mechanics plus special relativity. The result would be the so-called Bekenstein bound, which upper-bounds the number of bits in any spatial region by the product of the region’s radius and its energy content. I learned that there’s an interesting history here: Bekenstein originally deduced this bound using ingenious thought experiments involving black holes. Only later did people realize that the Bekenstein bound can be derived purely within QFT (see here and here for example)—in contrast to the holographic bound, which really is a statement about quantum gravity. (An early hint of this was that, while the holographic bound involves Newton’s gravitational constant G, the Bekenstein bound doesn’t.)

Thirdly, speaking of QFT, some readers might be struck by the fact that at no point in our 12-step program did we ever seem to need QFT machinery. Which is fortunate, because if we had needed it, I wouldn’t have been able to explain any of this! But here I have to confess that I cheated slightly. Recall step 4, which said that “if you know the rate at which a field varies across space, you can calculate the rate at which it varies across time.” It turns out that, in order to give that sentence a definite meaning, one uses the fact that in QFT, space and time derivatives in the Hamiltonian need to be related by a factor of c, since otherwise the Hamiltonian wouldn’t be Lorentz-invariant.

Fourthly, eagle-eyed readers might notice a loophole in the argument. Namely, we never upper-bounded how much information God could add to the world, via fields that are constant across all of spacetime. For example, there’s nothing to stop Her from creating a new scalar field that takes the same value everywhere in the universe—with that value, in suitable units, encoding 1050000 separate divine thoughts in its binary expansion. But OK, being constant, such a field would interact with nothing and affect no observations—so Occam’s Razor itches to slice it off, by rewriting the laws of physics in a simpler form where that field is absent. If you like, such a field would at most be a comment in the source code of the universe: it could be as long as the Great Programmer wanted it to be, but would have no observable effect on those of us living inside the program’s execution.

Of course, even before relativity and quantum mechanics, information had already been playing a surprisingly fleshy role in physics, through its appearance as entropy in 19th-century thermodynamics. Which leads to another puzzle. To a computer scientist, the concept of entropy, as the log of the number of microstates compatible with a given macrostate, seems clear enough, as does the intuition for why it should increase monotonically with time. Or at least, to whatever extent we’re confused about these matters, we’re no more confused than the physicists are!

But then why should this information-theoretic concept be so closely connected to tangible quantities like temperature, and pressure, and energy? From the mere assumption that a black hole has a nonzero entropy—that is, that it takes many bits to describe—how could Bekenstein and Hawking have possibly deduced that it also has a nonzero temperature? Or: if you put your finger into a tub of hot water, does the heat that you feel somehow reflect how many bits are needed to describe the water’s microstate?

Once again our skeptic pipes up: “but surely God could stuff as many additional bits as She wanted into the microstate of the hot water—for example, in degrees of freedom that are still unknown to physics—without the new bits having any effect on the water’s temperature.”

But we should’ve learned by now to doubt this sort of argument. There’s no general principle, in our universe, saying that you can hide as many bits as you want in a physical object, without those bits influencing the object’s observable properties. On the contrary, in case after case, our laws of physics seem to be intolerant of “wallflower bits,” which hide in a corner without talking to anyone. If a bit is there, the laws of physics want it to affect other nearby bits and be affected by them in turn.

In the case of thermodynamics, the assumption that does all the real work here is that of equidistribution. That is, whatever degrees of freedom might be available to your thermal system, your gas in a box or whatever, we assume that they’re all already “as randomized as they could possibly be,” subject to a few observed properties like temperature and volume and pressure. (At least, we assume that in classical thermodynamics. Non-equilibrium thermodynamics is a whole different can of worms, worms that don’t stay in equilibrium.) Crucially, we assume this despite the fact that we might not even know all the relevant degrees of freedom.

Why is this assumption justified? “Because experiment bears it out,” the physics teacher explains—but we can do better. The assumption is justified because, as long as the degrees of freedom that we’re talking about all interact with each other, they’ve already had plenty of time to equilibrate. And conversely, if a degree of freedom doesn’t interact with the stuff we’re observing—or with anything that interacts with the stuff we’re observing, etc.—well then, who cares about it anyway?

But now, because the microscopic laws of physics have the fundamental property of reversibility—that is, they never destroy information—a new bit has to go somewhere, and it can’t overwrite degrees of freedom that are already fully randomized. This is why, if you pump more bits of information into a tub of hot water, while keeping it at the same volume, the new bits have nowhere to go except into pushing up the energy. Now, there are often ways to push up the energy other than by raising the temperature—the concept of specific heat, in chemistry, is precisely about this—but if you need to stuff more bits into a substance, at the cost of raising its energy, certainly one of the obvious ways to do it is to describe a greater range of possible speeds for the water molecules. So since that can happen, by equidistribution it typically does happen, which means that the molecules move faster on average, and your finger feels the water get hotter.

In summary, our laws of physics are structured in such a way that even pure information often has “nowhere to hide”: if the bits are there at all in the abstract machinery of the world, then they’re forced to pipe up and have a measurable effect. And this is not a tautology, but comes about only because of nontrivial facts about special and general relativity, quantum mechanics, quantum field theory, and thermodynamics. And this is what I think people should mean when they say “information is physical.”

Anyway, if this was all obvious to you, I apologize for having wasted your time! But in my defense, it was never explained to me quite this way, nor was it sorted out in my head until recently—even though it seems like one of the most basic and general things one can possibly say about physics.

Endnotes. Thanks again to Daniel Harlow, not only for explaining the logic of the holographic bound to me but for several suggestions that improved this post.

Some readers might suspect circularity in the arguments we’ve made: are we merely saying that “any information that has observable physical consequences, has observable physical consequences”? No, it’s more than that. In all the examples I discussed, the magic was that we inserted certain information into our abstract mathematical description of the world, taking no care to ensure that the information’s presence would have any observable consequences whatsoever. But then the principles of quantum mechanics, quantum gravity, or thermodynamics forced the information to be detectable in very specific ways (namely, via the destruction of quantum interference, the warping of spacetime, or the generation of heat respectively).

Unrelated Update (June 6): It looks like the issues we’ve had with commenting have finally been fixed! Thanks so much to Christie Wright and others at WordPress Concierge Services for handling this. Let me know if you still have problems. In the meantime, I also stopped asking for commenters’ email addresses (many commenters filled that field with nonsense anyway). Oops, that ended up being a terrible idea, because it made commenting impossible! Back to how it was before.

Update (June 5): Erik Hoel was kind enough to write a 5-page response to this post (Word .docx format), and to give me permission to share it here. I might respond to various parts of it later. For now, though, I’ll simply say that I stand by what I wrote, and that requiring the macro-distribution to arise by marginalizing the micro-distribution still seems like the correct choice to me (and is what’s assumed in, e.g., the proof of the data processing inequality). But I invite readers to read my post along with Erik’s response, form their own opinions, and share them in the comments section.

In their new work, Hoel and others claim to make the amazing discovery that scientific reductionism is false—or, more precisely, that there can exist “causal information” in macroscopic systems, information relevant for predicting the systems’ future behavior, that’s not reducible to causal information about the systems’ microscopic building blocks. For more about what we’ll be discussing, see Hoel’s FQXi essay “Agent Above, Atom Below,” or better yet, his paper in Entropy, When the Map Is Better Than the Territory. Here’s the abstract of the Entropy paper:

The causal structure of any system can be analyzed at a multitude of spatial and temporal scales. It has long been thought that while higher scale (macro) descriptions may be useful to observers, they are at best a compressed description and at worse leave out critical information and causal relationships. However, recent research applying information theory to causal analysis has shown that the causal structure of some systems can actually come into focus and be more informative at a macroscale. That is, a macroscale description of a system (a map) can be more informative than a fully detailed microscale description of the system (the territory). This has been called “causal emergence.” While causal emergence may at first seem counterintuitive, this paper grounds the phenomenon in a classic concept from information theory: Shannon’s discovery of the channel capacity. I argue that systems have a particular causal capacity, and that different descriptions of those systems take advantage of that capacity to various degrees. For some systems, only macroscale descriptions use the full causal capacity. These macroscales can either be coarse-grains, or may leave variables and states out of the model (exogenous, or “black boxed”) in various ways, which can improve the efficacy and informativeness via the same mathematical principles of how error-correcting codes take advantage of an information channel’s capacity. The causal capacity of a system can approach the channel capacity as more and different kinds of macroscales are considered. Ultimately, this provides a general framework for understanding how the causal structure of some systems cannot be fully captured by even the most detailed microscale description.

Anyway, Wolchover’s popular article quoted various researchers praising the theory of causal emergence, as well as a single inexplicably curmudgeonly skeptic—some guy who sounded like he was so off his game (or maybe just bored with debates about ‘reductionism’ versus ’emergence’?), that he couldn’t even be bothered to engage the details of what he was supposed to be commenting on.

Hoel’s ideas do not impress Scott Aaronson, a theoretical computer scientist at the University of Texas, Austin. He says causal emergence isn’t radical in its basic premise. After reading Hoel’s recent essay for the Foundational Questions Institute, “Agent Above, Atom Below” (the one that featured Romeo and Juliet), Aaronson said, “It was hard for me to find anything in the essay that the world’s most orthodox reductionist would disagree with. Yes, of course you want to pass to higher abstraction layers in order to make predictions, and to tell causal stories that are predictively useful — and the essay explains some of the reasons why.”

After the Quanta piece came out, Sean Carroll tweeted approvingly about the above paragraph, calling me a “voice of reason [yes, Sean; have I ever not been?], slapping down the idea that emergent higher levels have spooky causal powers.” Then Sean, in turn, was criticized for that remark by Hoel and others.

Hoel in particular raised a reasonable-sounding question. Namely, in my “curmudgeon paragraph” from Wolchover’s article, I claimed that the notion of “causal emergence,” or causality at the macro-scale, says nothing fundamentally new. Instead it simply reiterates the usual worldview of science, according to which

the universe is ultimately made of quantum fields evolving by some Hamiltonian, but

if someone asks (say) “why has air travel in the US gotten so terrible?”, a useful answer is going to talk about politics or psychology or economics or history rather than the movements of quarks and leptons.

But then, Hoel asks, if there’s nothing here for the world’s most orthodox reductionist to disagree with, then how do we find Carroll and other reductionists … err, disagreeing?

I think this dilemma is actually not hard to resolve. Faced with a claim about “causation at higher levels,” what reductionists disagree with is not the object-level claim that such causation exists (I scratched my nose because it itched, not because of the Standard Model of elementary particles). Rather, they disagree with the meta-level claim that there’s anything shocking about such causation, anything that poses a special difficulty for the reductionist worldview that physics has held for centuries. I.e., they consider it true both that

my nose is made of subatomic particles, and its behavior is in principle fully determined (at least probabilistically) by the quantum state of those particles together with the laws governing them, and

my nose itched.

At least if we leave the hard problem of consciousness out of it—that’s a separate debate—there seems to be no reason to imagine a contradiction between 1 and 2 that needs to be resolved, but “only” a vast network of intervening mechanisms to be elucidated. So, this is how it is that reductionists can find anti-reductionist claims to be both wrong and vacuously correct at the same time.

(Incidentally, yes, quantum entanglement provides an obvious sense in which “the whole is more than the sum of its parts,” but even in quantum mechanics, the whole isn’t more than the density matrix, which is still a huge array of numbers evolving by an equation, just different numbers than one would’ve thought a priori. For that reason, it’s not obvious what relevance, if any, QM has to reductionism versus anti-reductionism. In any case, QM is not what Hoel invokes in his causal emergence theory.)

From reading the philosophical parts of Hoel’s papers, it was clear to me that some remarks like the above might help ward off the forehead-banging confusions that these discussions inevitably provoke. So standard-issue crustiness is what I offered Natalie Wolchover when she asked me, not having time on short notice to go through the technical arguments.

But of course this still leaves the question: what is in the mathematical part of Hoel’s Entropy paper? What exactly is it that the advocates of causal emergence claim provides a new argument against reductionism?

To answer that question, yesterday I (finally) read the Entropy paper all the way through.

Much like Tononi’s integrated information theory was built around a numerical measure called Φ, causal emergence is built around a different numerical quantity, this one supposed to measure the amount of “causal information” at a particular scale. The measure is called effective information or EI, and it’s basically the mutual information between a system’s initial state sI and its final state sF, assuming a uniform distribution over sI. Much like with Φ in IIT, computations of this EI are then used as the basis for wide-ranging philosophical claims—even though EI, like Φ, has aspects that could be criticized as arbitrary, and as not obviously connected with what we’re trying to understand.

Once again like with Φ, one of those assumptions is that of a uniform distribution over one of the variables, sI, whose relatedness we’re trying to measure. In my IIT post, I remarked on that assumption, but I didn’t harp on it, since I didn’t see that it did serious harm, and in any case my central objection to Φ would hold regardless of which distribution we chose. With causal emergence, by contrast, this uniformity assumption turns out to be the key to everything.

For here is the argument from the Entropy paper, for the existence of macroscopic causality that’s not reducible to causality in the underlying components. Suppose I have a system with 8 possible states (called “microstates”), which I label 1 through 8. And suppose the system evolves as follows: if it starts out in states 1 through 7, then it goes to state 1. If, on the other hand, it starts in state 8, then it stays in state 8. In such a case, it seems reasonable to “coarse-grain” the system, by lumping together initial states 1 through 7 into a single “macrostate,” call it A, and letting the initial state 8 comprise a second macrostate, call it B.

We now ask: how much information does knowing the system’s initial state tell you about its final state? If we’re talking about microstates, and we let the system start out in a uniform distribution over microstates 1 through 8, then 7/8 of the time the system goes to state 1. So there’s just not much information about the final state to be predicted—specifically, only 7/8×log2(8/7) + 1/8×log2(8) ≈ 0.54 bits of entropy—which, in this case, is also the mutual information between the initial and final microstates. If, on the other hand, we’re talking about macrostates, and we let the system start in a uniform distribution over macrostates A and B, then A goes to A and B goes to B. So knowing the initial macrostate gives us 1 full bit of information about the final state, which is more than the ~0.54 bits that looking at the microstate gave us! Ergo reductionism is false.

Once the argument is spelled out, it’s clear that the entire thing boils down to, how shall I put this, a normalization issue. That is: we insist on the uniform distribution over microstates when calculating microscopic EI, and we also insist on the uniform distribution over macrostates when calculating macroscopic EI, and we ignore the fact that the uniform distribution over microstates gives rise to a non-uniform distribution over macrostates, because some macrostates can be formed in more ways than others. If we fixed this, demanding that the two distributions be compatible with each other, we’d immediately find that, surprise, knowing the complete initial microstate of a system always gives you at least as much power to predict the system’s future as knowing a macroscopic approximation to that state. (How could it not? For given the microstate, we could in principle compute the macroscopic approximation for ourselves, but not vice versa.)

The closest the paper comes to acknowledging the problem—i.e., that it’s all just a normalization trick—seems to be the following paragraph in the discussion section:

Another possible objection to causal emergence is that it is not natural but rather enforced upon a system via an experimenter’s application of an intervention distribution, that is, from using macro-interventions. For formalization purposes, it is the experimenter who is the source of the intervention distribution, which reveals a causal structure that already exists. Additionally, nature itself may intervene upon a system with statistical regularities, just like an intervention distribution. Some of these naturally occurring input distributions may have a viable interpretation as a macroscale causal model (such as being equal to Hmax [the maximum entropy] at some particular macroscale). In this sense, some systems may function over their inputs and outputs at a microscale or macroscale, depending on their own causal capacity and the probability distribution of some natural source of driving input.

As far as I understand it, this paragraph is saying that, for all we know, something could give rise to a uniform distribution over macrostates, so therefore that’s a valid thing to look at, even if it’s not what we get by taking a uniform distribution over microstates and then coarse-graining it. Well, OK, but unknown interventions could give rise to many other distributions over macrostates as well. In any case, if we’re directly comparing causal information at the microscale against causal information at the macroscale, it still seems reasonable to me to demand that in the comparison, the macro-distribution arise by coarse-graining the micro one. But in that case, the entire argument collapses.

Despite everything I said above, the real purpose of this post is to announce that I’ve changed my mind. I now believe that, while Hoel’s argument might be unsatisfactory, the conclusion is fundamentally correct: scientific reductionism is false. There is higher-level causation in our universe, and it’s 100% genuine, not just a verbal sleight-of-hand. In particular, there are causal forces that can only be understood in terms of human desires and goals, and not in terms of subatomic particles blindly bouncing around.

So what caused such a dramatic conversion?

By 2015, after decades of research and diplomacy and activism and struggle, 196 nations had finally agreed to limit their carbon dioxide emissions—every nation on earth besides Syria and Nicaragua, and Nicaragua only because it thought the agreement didn’t go far enough. The human race had thereby started to carve out some sort of future for itself, one in which the oceans might rise slowly enough that we could adapt, and maybe buy enough time until new technologies were invented that changed the outlook. Of course the Paris agreement fell far short of what was needed, but it was a start, something to build on in the coming decades. Even in the US, long the hotbed of intransigence and denial on this issue, 69% of the public supported joining the Paris agreement, compared to a mere 13% who opposed. Clean energy was getting cheaper by the year. Most of the US’s largest corporations, including Google, Microsoft, Apple, Intel, Mars, PG&E, and ExxonMobil—ExxonMobil, for godsakes—vocally supported staying in the agreement and working to cut their own carbon footprints. All in all, there was reason to be cautiously optimistic that children born today wouldn’t live to curse their parents for having brought them into a world so close to collapse.

In order to unravel all this, in order to steer the heavy ship of destiny off the path toward averting the crisis and toward the path of existential despair, a huge number of unlikely events would need to happen in succession, as if propelled by some evil supernatural force.

Like what? I dunno, maybe a fascist demagogue would take over the United States on a campaign based on willful cruelty, on digging up and burning dirty fuels just because and even if it made zero economic sense, just for the fun of sticking it to liberals, or because of the urgent need to save the US coal industry, which employs fewer people than Arby’s. Such a demagogue would have no chance of getting elected, you say?

So let’s suppose he’s up against a historically unpopular opponent. Let’s suppose that even then, he still loses the popular vote, but somehow ekes out an Electoral College win. Maybe he gets crucial help in winning the election from a hostile foreign power—and for some reason, pro-American nationalists are totally OK with that, even cheer it. Even then, we’d still probably need a string of additional absurd coincidences. Like, I dunno, maybe the fascist’s opponent has an aide who used to be married to a guy who likes sending lewd photos to minors, and investigating that guy leads the FBI to some emails that ultimately turn out to mean nothing whatsoever, but that the media hyperventilate about precisely in time to cause just enough people to vote to bring the fascist to power, thereby bringing about the end of the world. Something like that.

It’s kind of like, you know that thing where the small population in Europe that produced Einstein and von Neumann and Erdös and Ulam and Tarski and von Karman and Polya was systematically exterminated (along with millions of other innocents) soon after it started producing such people, and the world still hasn’t fully recovered? How many things needed to go wrong for that to happen? Obviously you needed Hitler to be born, and to survive the trenches and assassination plots; and Hindenburg to make the fateful decision to give Hitler power. But beyond that, the world had to sleep as Germany rebuilt its military; every last country had to turn away refugees; the UK had to shut down Jewish immigration to Palestine at exactly the right time; newspapers had to bury the story; government record-keeping had to have advanced just to the point that rounding up millions for mass murder was (barely) logistically possible; and finally, the war had to continue long enough for nearly every European country to have just enough time to ship its Jews to their deaths, before the Allies showed up to liberate mostly the ashes.

In my view, these simply aren’t the sort of outcomes that you expect from atoms blindly interacting according to the laws of physics. These are, instead, the signatures of higher-level causation—and specifically, of a teleological force that operates in our universe to make it distinctively cruel and horrible.

Admittedly, I don’t claim to know the exact mechanism of the higher-level causation. Maybe, as the physicist Yakir Aharonov has advocated, our universe has not only a special, low-entropy initial state at the Big Bang, but also a “postselected final state,” toward which the outcomes of quantum measurements get mysteriously “pulled”—an effect that might show up in experiments as ever-so-slight deviations from the Born rule. And because of the postselected final state, even if the human race naïvely had only (say) a one-in-thousand chance of killing itself off, even if the paths to its destruction all involved some improbable absurdity, like an orange clown showing up from nowhere—nevertheless, the orange clown would show up. Alternatively, maybe the higher-level causation unfolds through subtle correlations in the universe’s initial state, along the lines I sketched in my 2013 essay The Ghost in the Quantum Turing Machine. Or maybe Erik Hoel is right after all, and it all comes down to normalization: if we looked at the uniform distribution over macrostates rather than over microstates, we’d discover that orange clowns destroying the world predominated. Whatever the details, though, I think it can no longer be doubted that we live, not in the coldly impersonal universe that physics posited for centuries, but instead in a tragicomically evil one.

I call my theory reverse Hollywoodism, because it holds that the real world has the inverse of the typical Hollywood movie’s narrative arc. Again and again, what we observe is that the forces of good have every possible advantage, from money to knowledge to overwhelming numerical superiority. Yet somehow good still fumbles. Somehow a string of improbable coincidences, or a black swan or an orange Hitler, show up at the last moment to let horribleness eke out a last-minute victory, as if the world itself had been rooting for horribleness all along. That’s our universe.

I’m fine if you don’t believe this theory: maybe you’re congenitally more optimistic than I am (in which case, more power to you); maybe the full weight of our universe’s freakish awfulness doesn’t bear down on you as it does on me. But I hope you’ll concede that, if nothing else, this theory is a genuinely non-reductionist one.

Yesterday Ryan Mandelbaum, at Gizmodo, posted a decidedly tongue-in-cheek piece about whether or not the universe is a computer simulation. (The piece was filed under the category “LOL.”)

The immediate impetus for Mandelbaum’s piece was a blog post by Sabine Hossenfelder, a physicist who will likely be familiar to regulars here in the nerdosphere. In her post, Sabine vents about the simulation speculations of philosophers like Nick Bostrom. She writes:

Proclaiming that “the programmer did it” doesn’t only not explain anything – it teleports us back to the age of mythology. The simulation hypothesis annoys me because it intrudes on the terrain of physicists. It’s a bold claim about the laws of nature that however doesn’t pay any attention to what we know about the laws of nature.

After hammering home that point, Sabine goes further, and says that the simulation hypothesis is almost ruled out, by (for example) the fact that our universe is Lorentz-invariant, and a simulation of our world by a discrete lattice of bits won’t reproduce Lorentz-invariance or other continuous symmetries.

In writing his post, Ryan Mandelbaum interviewed two people: Sabine and me.

I basically told Ryan that I agree with Sabine insofar as she argues that the simulation hypothesis is lazy—that it doesn’t pay its rent by doing real explanatory work, doesn’t even engage much with any of the deep things we’ve learned about the physical world—and disagree insofar as she argues that the simulation hypothesis faces some special difficulty because of Lorentz-invariance or other continuous phenomena in known physics. In short: blame it for being unfalsifiable rather than for being falsified!

Indeed, to whatever extent we believe the Bekenstein bound—and even more pointedly, to whatever extent we think the AdS/CFT correspondence says something about reality—we believe that in quantum gravity, any bounded physical system (with a short-wavelength cutoff, yada yada) lives in a Hilbert space of a finite number of qubits, perhaps ~1069 qubits per square meter of surface area. And as a corollary, if the cosmological constant is indeed constant (so that galaxies more than ~20 billion light years away are receding from us faster than light), then our entire observable universe can be described as a system of ~10122 qubits. The qubits would in some sense be the fundamental reality, from which Lorentz-invariant spacetime and all the rest would need to be recovered as low-energy effective descriptions. (I hasten to add: there’s of course nothing special about qubits here, any more than there is about bits in classical computation, compared to some other unit of information—nothing that says the Hilbert space dimension has to be a power of 2 or anything silly like that.) Anyway, this would mean that our observable universe could be simulated by a quantum computer—or even for that matter by a classical computer, to high precision, using a mere ~210^122 time steps.

Sabine might respond that AdS/CFT and other quantum gravity ideas are mere theoretical speculations, not solid and established like special relativity. But crucially, if you believe that the observable universe couldn’t be simulated by a computer even in principle—that it has no mapping to any system of bits or qubits—then at some point the speculative shoe shifts to the other foot. The question becomes: do you reject the Church-Turing Thesis? Or, what amounts to the same thing: do you believe, like Roger Penrose, that it’s possible to build devices in nature that solve the halting problem or other uncomputable problems? If so, how? But if not, then how exactly does the universe avoid being computational, in the broad sense of the term?

I’d write more, but by coincidence, right now I’m at an It from Qubit meeting at Stanford, where everyone is talking about how to map quantum theories of gravity to quantum circuits acting on finite sets of qubits, and the questions in quantum circuit complexity that are thereby raised. It’s tremendously exciting—the mixture of attendees is among the most stimulating I’ve ever encountered, from Lenny Susskind and Don Page and Daniel Harlow to Umesh Vazirani and Dorit Aharonov and Mario Szegedy to Google’s Sergey Brin. But it should surprise no one that, amid all the discussion of computation and fundamental physics, the question of whether the universe “really” “is” a simulation has barely come up. Why would it, when there are so many more fruitful things to ask? All I can say with confidence is that, if our world is a simulation, then whoever is simulating it (God, or a bored teenager in the metaverse) seems to have a clear preference for the 2-norm over the 1-norm, and for the complex numbers over the reals.

Happy New Year, everyone! I tripped over a well-concealed hole and sprained my ankle while carrying my daughter across the grass at Austin’s New Years festival, so am now ringing in 2017 lying in bed immobilized, which somehow seems appropriate. At least Lily is fine, and at least being bedridden gives me ample opportunity to blog.

Another year, another annual Edge question, with its opportunity for hundreds of scientists and intellectuals (including yours truly) to pontificate, often about why their own field of study is the source of the most important insights and challenges facing humanity. This year’s question was:

With the example given of Richard Dawkins’s “meme,” which jumped into the general vernacular, becoming a meme itself.

My entry, about the notion of “state” (yeah, I tried to focus on the basics), is here.

This year’s question presented a particular challenge, which scientists writing for a broad audience might not have faced for generations. Namely: to what extent, if any, should your writing acknowledge the dark shadow of recent events? Does the Putinization of the United States render your little pet debates and hobbyhorses irrelevant? Or is the most defiant thing you can do to ignore the unfolding catastrophe, to continue building your intellectual sandcastle even as the tidal wave of populist hatred nears?

In any case, the instructions from Edge were clear: ignore politics. Focus on the eternal. But people interpreted that injunction differently.

One of my first ideas was to write about the Second Law of Thermodynamics, and to muse about how one of humanity’s tragic flaws is to take for granted the gargantuan effort needed to create and maintain even little temporary pockets of order. Again and again, people imagine that, if their local pocket of order isn’t working how they want, then they should smash it to pieces, since while admittedly that might make things even worse, there’s also at least 50/50 odds that they’ll magically improve. In reasoning thus, people fail to appreciate just how exponentially more numerous are the paths downhill, into barbarism and chaos, than are the few paths further up. So thrashing about randomly, with no knowledge or understanding, is statistically certain to make things worse: on this point thermodynamics, common sense, and human history are all in total agreement. The implications of these musings for the present would be left as exercises for the reader.

Anyway, I was then pleased when, in a case of convergent evolution, my friend and hero Steven Pinker wrote exactly that essay, so I didn’t need to.

There are many other essays that are worth a read, some of which allude to recent events but the majority of which don’t. Let me mention a few.

Nicholas Humphrey on referential opacity: while I didn’t know the term before, this is precisely the reason why, even if P=NP, that still wouldn’t imply PA=NPA for all oracles A.

Donald Hoffman on the holographic principle. For the point he wanted to make, about the mismatch between our intuitions and the physical world, it seems to me that Hoffman could’ve picked pretty much anything in physics, from Galileo and Newton onward. What’s new about holography?

Jerry Coyne on determinism. Coyne, who’s written many things I admire, here offers his version of an old argument that I tear my hair out every time I read. There’s no free will, Coyne says, and therefore we should treat criminals more lightly, e.g. by eschewing harsh punishments in favor of rehabilitation. Following tradition, Coyne never engages the obvious reply, which is: “sorry, to whom were you addressing that argument? To me, the jailer? To the judge? The jury? Voters? Were you addressing us as moral agents, for whom the concept of ‘should’ is relevant? Then why shouldn’t we address the criminals the same way?”

Michael Gazzaniga on “The Schnitt.” Yes, it’s possible that things like the hard problem of consciousness, or the measurement problem in quantum mechanics, will never have a satisfactory resolution. But even if so, building a complicated verbal edifice whose sole purpose is to tell people not even to look for a solution, to be satisfied with two “non-overlapping magisteria” and a lack of any explanation for how to reconcile them, never struck me as a substantive contribution to knowledge. It wasn’t when Niels Bohr did it, and it’s not when someone today does it either.

I had a related quibble with Amanda Gefter’s piece on “enactivism”: the view she takes as her starting point, that “physics proves there’s no third-person view of the world,” is controversial to put it mildly among those who know the relevant physics. (And even if we granted that view, surely a third-person perspective exists for the quasi-Newtonian world in which we evolved, and that’s relevant for the cognitive science questions Gefter then discusses.)

Thomas Bass on information pathology. Bass obliquely discusses the propaganda, conspiracy theories, social-media echo chambers, and unchallenged lies that helped fuel Trump’s rise. He then locates the source of the problem in Shannon’s information theory (!), which told us how to quantify information, but failed to address questions about the information’s meaning or relevance. To me, this is almost exactly like blaming arithmetic because it only tells you how to add numbers, without caring whether they’re numbers of rescued orphans or numbers of bombs. Arithmetic is fine; the problem is with us.

In his piece on “number sense,” Keith Devlin argues that the teaching of “rigid, rule-based” math has been rendered obsolete by computers, leaving only the need to teach high-level conceptual understanding. I partly agree and partly disagree, with the disagreement coming from firsthand knowledge of just how badly that lofty idea gets beaten to mush once it filters down to the grade-school level. I would say that the basic function of math education is to teach clarity of thought: does this statement hold for all positive integers, or not? Not how do you feel about it, but does it hold? If it holds, can you prove it? What other statements would it follow from? If it doesn’t hold, can you give a counterexample? (Incidentally, there are plenty of questions of this type for which humans still outperform the best available software!) Admittedly, pencil-and-paper arithmetic is both boring and useless—but if you never mastered anything like it, then you certainly wouldn’t be ready for the concept of an algorithm, or for asking higher-level questions about algorithms.

Daniel Hook on PT-symmetric quantum mechanics. As far as I understand, PT-symmetric Hamiltonians are equivalent to ordinary Hermitian ones under similarity transformations. So this is a mathematical trick, perhaps a useful one—but it’s extremely misleading to talk about it as if it were a new physical theory that differed from quantum mechanics.

Jared Diamond extols the virtues of common sense, of which there are indeed many—but alas, his example is that if a mathematical proof leads to a conclusion that your common sense tells you is wrong, then you shouldn’t waste time looking for the exact mistake. Sometimes that’s good advice, but it’s pretty terrible applied to Goodstein’s Theorem, the muddy children puzzle, the strategy-stealing argument for Go, or anything else that genuinely is shocking until your common sense expands to accommodate it. Math, like science in general, is a constant dialogue between formal methods and common sense, where sometimes it’s one that needs to get with the program and sometimes it’s the other.

Hans Halvorson on matter. I take issue with Halvorson’s claim that quantum mechanics had to be discarded in favor of quantum field theory, because QM was inconsistent with special relativity. It seems much better to say: the thing that conflicts with special relativity, and that quantum field theory superseded, was a particular application of quantum mechanics, involving wavefunctions of N particles moving around in a non-relativistic space. The general principles of QM—unit vectors in complex Hilbert space, unitary evolution, the Born rule, etc.—survived the transition to QFT without the slightest change.

The following are the after-dinner remarks that I delivered at QCRYPT’2016, the premier quantum cryptography conference, on Thursday Sep. 15 in Washington DC. You could compare to my after-dinner remarks at QIP’2006 to see how much I’ve “”matured”” since then. Thanks so much to Yi-Kai Liu and the other organizers for inviting me and for putting on a really fantastic conference.

It’s wonderful to be here at QCRYPT among so many friends—this is the first significant conference I’ve attended since I moved from MIT to Texas. I do, however, need to register a complaint with the organizers, which is: why wasn’t I allowed to bring my concealed firearm to the conference? You know, down in Texas, we don’t look too kindly on you academic elitists in Washington DC telling us what to do, who we can and can’t shoot and so forth. Don’t mess with Texas! As you might’ve heard, many of us Texans even support a big, beautiful, physical wall being built along our border with Mexico. Personally, though, I don’t think the wall proposal goes far enough. Forget about illegal immigration and smuggling: I don’t even want Americans and Mexicans to be able to win the CHSH game with probability exceeding 3/4. Do any of you know what kind of wall could prevent that? Maybe a metaphysical wall.

OK, but that’s not what I wanted to talk about. When Yi-Kai asked me to give an after-dinner talk, I wasn’t sure whether to try to say something actually relevant to quantum cryptography or just make jokes. So I’ll do something in between: I’ll tell you about research directions in quantum cryptography that are also jokes.

The subject of this talk is a deep theorem that stands as one of the crowning achievements of our field. I refer, of course, to the No-Cloning Theorem. Almost everything we’re talking about at this conference, from QKD onwards, is based in some way on quantum states being unclonable. If you read Stephen Wiesner’s paper from 1968, which founded quantum cryptography, the No-Cloning Theorem already played a central role—although Wiesner didn’t call it that. By the way, here’s my #1 piece of research advice to the students in the audience: if you want to become immortal, just find some fact that everyone already knows and give it a name!

I’d like to pose the question: why should our universe be governed by physical laws that make the No-Cloning Theorem true? I mean, it’s possible that there’s some other reason for our universe to be quantum-mechanical, and No-Cloning is just a byproduct of that. No-Cloning would then be like the armpit of quantum mechanics: not there because it does anything useful, but just because there’s gotta be something under your arms.

OK, but No-Cloning feels really fundamental. One of my early memories is when I was 5 years old or so, and utterly transfixed by my dad’s home fax machine, one of those crappy 1980s fax machines with wax paper. I kept thinking about it: is it really true that a piece of paper gets transmaterialized, sent through a wire, and reconstituted at the other location? Could I have been that wrong about how the universe works? Until finally I got it—and once you get it, it’s hard even to recapture your original confusion, because it becomes so obvious that the world is made not of stuff but of copyable bits of information. “Information wants to be free!”

The No-Cloning Theorem represents nothing less than a partial return to the view of the world that I had before I was five. It says that quantum information doesn’t want to be free: it wants to be private. There is, it turns out, a kind of information that’s tied to a particular place, or set of places. It can be moved around, or even teleported, but it can’t be copied the way a fax machine copies bits.

So I think it’s worth at least entertaining the possibility that we don’t have No-Cloning because of quantum mechanics; we have quantum mechanics because of No-Cloning—or because quantum mechanics is the simplest, most elegant theory that has unclonability as a core principle. But if so, that just pushes the question back to: why should unclonability be a core principle of physics?

Quantum Key Distribution

A first suggestion about this question came from Gilles Brassard, who’s here. Years ago, I attended a talk by Gilles in which he speculated that the laws of quantum mechanics are what they are because Quantum Key Distribution (QKD) has to be possible, while bit commitment has to be impossible. If true, that would be awesome for the people at this conference. It would mean that, far from being this exotic competitor to RSA and Diffie-Hellman that’s distance-limited and bandwidth-limited and has a tiny market share right now, QKD would be the entire reason why the universe is as it is! Or maybe what this really amounts to is an appeal to the Anthropic Principle. Like, if QKD hadn’t been possible, then we wouldn’t be here at QCRYPT to talk about it.

Quantum Money

But maybe we should search more broadly for the reasons why our laws of physics satisfy a No-Cloning Theorem. Wiesner’s paper sort of hinted at QKD, but the main thing it had was a scheme for unforgeable quantum money. This is one of the most direct uses imaginable for the No-Cloning Theorem: to store economic value in something that it’s physically impossible to copy. So maybe that’s the reason for No-Cloning: because God wanted us to have e-commerce, and didn’t want us to have to bother with blockchains (and certainly not with credit card numbers).

The central difficulty with quantum money is: how do you authenticate a bill as genuine? (OK, fine, there’s also the dificulty of how to keep a bill coherent in your wallet for more than a microsecond or whatever. But we’ll leave that for the engineers.)

In Wiesner’s original scheme, he solved the authentication problem by saying that, whenever you want to verify a quantum bill, you bring it back to the bank that printed it. The bank then looks up the bill’s classical serial number in a giant database, which tells the bank in which basis to measure each of the bill’s qubits.

With this system, you can actually get information-theoretic security against counterfeiting. OK, but the fact that you have to bring a bill to the bank to be verified negates much of the advantage of quantum money in the first place. If you’re going to keep involving a bank, then why not just use a credit card?

That’s why over the past decade, some of us have been working on public-key quantum money: that is, quantum money that anyone can verify. For this kind of quantum money, it’s easy to see that the No-Cloning Theorem is no longer enough: you also need some cryptographic assumption. But OK, we can consider that. In recent years, we’ve achieved glory by proposing a huge variety of public-key quantum money schemes—and we’ve achieved even greater glory by breaking almost all of them!

After a while, there were basically two schemes left standing: one based on knot theory by Ed Farhi, Peter Shor, et al. That one has been proven to be secure under the assumption that it can’t be broken. The second scheme, which Paul Christiano and I proposed in 2012, is based on hidden subspaces encoded by multivariate polynomials. For our scheme, Paul and I were able to do better than Farhi et al.: we gave a security reduction. That is, we proved that our quantum money scheme is secure, unless there’s a polynomial-time quantum algorithm to find hidden subspaces encoded by low-degree multivariate polynomials (yadda yadda, you can look up the details) with much greater success probability than we thought possible.

Today, the situation is that my and Paul’s security proof remains completely valid, but meanwhile, our money is completely insecure! Our reduction means the opposite of what we thought it did. There is a break of our quantum money scheme, and as a consequence, there’s also a quantum algorithm to find large subspaces hidden by low-degree polynomials with much better success probability than we’d thought. What happened was that first, some French algebraic cryptanalysts—Faugere, Pena, I can’t pronounce their names—used Gröbner bases to break the noiseless version of scheme, in classical polynomial time. So I thought, phew! At least I had acceded when Paul insisted that we also include a noisy version of the scheme. But later, Paul noticed that there’s a quantum reduction from the problem of breaking our noisy scheme to the problem of breaking the noiseless one, so the former is broken as well.

I’m choosing to spin this positively: “we used quantum money to discover a striking new quantum algorithm for finding subspaces hidden by low-degree polynomials. Err, yes, that’s exactly what we did.”

But, bottom line, until we manage to invent a better public-key quantum money scheme, or otherwise sort this out, I don’t think we’re entitled to claim that God put unclonability into our universe in order for quantum money to be possible.

Copy-Protected Quantum Software

So if not money, then what about its cousin, copy-protected software—could that be why No-Cloning holds? By copy-protected quantum software, I just mean a quantum state that, if you feed it into your quantum computer, lets you evaluate some Boolean function on any input of your choice, but that doesn’t let you efficiently prepare more states that let the same function be evaluated. I think this is important as one of the preeminent evil applications of quantum information. Why should nuclear physicists and genetic engineers get a monopoly on the evil stuff?

OK, but is copy-protected quantum software even possible? The first worry you might have is that, yeah, maybe it’s possible, but then every time you wanted to run the quantum program, you’d have to make a measurement that destroyed it. So then you’d have to go back and buy a new copy of the program for the next run, and so on. Of course, to the software company, this would presumably be a feature rather than a bug!

But as it turns out, there’s a fact many of you know—sometimes called the “Gentle Measurement Lemma,” other times the “Almost As Good As New Lemma”—which says that, as long as the outcome of your measurement on a quantum state could be predicted almost with certainty given knowledge of the state, the measurement can be implemented in such a way that it hardly damages the state at all. This tells us that, if quantum money, copy-protected quantum software, and the other things we’re talking about are possible at all, then they can also be made reusable. I summarize the principle as: “if rockets, then space shuttles.”

Much like with quantum money, one can show that, relative to a suitable oracle, it’s possible to quantumly copy-protect any efficiently computable function—or rather, any function that’s hard to learn from its input/output behavior. Indeed, the implementation can be not only copy-protected but also obfuscated, so that the user learns nothing besides the input/output behavior. As Bill Fefferman pointed out in his talk this morning, the No-Cloning Theorem lets us bypass Barak et al.’s famous result on the impossibility of obfuscation, because their impossibility proof assumed the ability to copy the obfuscated program.

Of course, what we really care about is whether quantum copy-protection is possible in the real world, with no oracle. I was able to give candidate implementations of quantum copy-protection for extremely special functions, like one that just checks the validity of a password. In the general case—that is, for arbitrary programs—Paul Christiano has a beautiful proposal for how to do it, which builds on our hidden-subspace money scheme. Unfortunately, since our money scheme is currently in the shop being repaired, it’s probably premature to think about the security of the much more complicated copy-protection scheme! But these are wonderful open problems, and I encourage any of you to come and scoop us. Once we know whether uncopyable quantum software is possible at all, we could then debate whether it’s the “reason” for our universe to have unclonability as a core principle.

Unclonable Proofs and Advice

Along the same lines, I can’t resist mentioning some favorite research directions, which some enterprising student here could totally turn into a talk at next year’s QCRYPT.

Firstly, what can we say about clonable versus unclonable quantum proofs—that is, QMA witness states? In other words: for which problems in QMA can we ensure that there’s an accepting witness that lets you efficiently create as many additional accepting witnesses as you want? (I mean, besides the QCMA problems, the ones that have short classical witnesses?) For which problems in QMA can we ensure that there’s an accepting witness that doesn’t let you efficiently create any additional accepting witnesses? I do have a few observations about these questions—ask me if you’re interested—but on the whole, I believe almost anything one can ask about them remains open.

Admittedly, it’s not clear how much use an unclonable proof would be. Like, imagine a quantum state that encoded a proof of the Riemann Hypothesis, and which you would keep in your bedroom, in a glass orb on your nightstand or something. And whenever you felt your doubts about the Riemann Hypothesis resurfacing, you’d take the state out of its orb and measure it again to reassure yourself of RH’s truth. You’d be like, “my preciousssss!” And no one else could copy your state and thereby gain the same Riemann-faith-restoring powers that you had. I dunno, I probably won’t hawk this application in a DARPA grant.

Similarly, one can ask about clonable versus unclonable quantum advice states—that is, initial states that are given to you to boost your computational power beyond that of an ordinary quantum computer. And that’s also a fascinating open problem.

OK, but maybe none of this quite gets at why our universe has unclonability. And this is an after-dinner talk, so do you want me to get to the really crazy stuff? Yes?

Self-Referential Paradoxes

OK! What if unclonability is our universe’s way around the paradoxes of self-reference, like the unsolvability of the halting problem and Gödel’s Incompleteness Theorem? Allow me to explain what I mean.

In kindergarten or wherever, we all learn Turing’s proof that there’s no computer program to solve the halting problem. But what isn’t usually stressed is that that proof actually does more than advertised. If someone hands you a program that they claim solves the halting problem, Turing doesn’t merely tell you that that person is wrong—rather, he shows you exactly how to expose the person as a jackass, by constructing an example input on which their program fails. All you do is, you take their claimed halt-decider, modify it in some simple way, and then feed the result back to the halt-decider as input. You thereby create a situation where, if your program halts given its own code as input, then it must run forever, and if it runs forever then it halts. “WHOOOOSH!” [head-exploding gesture]

OK, but now imagine that the program someone hands you, which they claim solves the halting problem, is a quantum program. That is, it’s a quantum state, which you measure in some basis depending on the program you’re interested in, in order to decide whether that program halts. Well, the truth is, this quantum program still can’t work to solve the halting problem. After all, there’s some classical program that simulates the quantum one, albeit less efficiently, and we already know that the classical program can’t work.

But now consider the question: how would you actually produce an example input on which this quantum program failed to solve the halting problem? Like, suppose the program worked on every input you tried. Then ultimately, to produce a counterexample, you might need to follow Turing’s proof and make a copy of the claimed quantum halt-decider. But then, of course, you’d run up against the No-Cloning Theorem!

So we seem to arrive at the conclusion that, while of course there’s no quantum program to solve the halting problem, there might be a quantum program for which no one could explicitly refute that it solved the halting problem, by giving a counterexample.

I was pretty excited about this observation for a day or two, until I noticed the following. Let’s suppose your quantum program that allegedly solves the halting problem has n qubits. Then it’s possible to prove that the program can’t possibly be used to compute more than, say, 2n bits of Chaitin’s constant Ω, which is the probability that a random program halts. OK, but if we had an actual oracle for the halting problem, we could use it to compute as many bits of Ω as we wanted. So, suppose I treated my quantum program as if it were an oracle for the halting problem, and I used it to compute the first 2n bits of Ω. Then I would know that, assuming the truth of quantum mechanics, the program must have made a mistake somewhere. There would still be something weird, which is that I wouldn’t know on which input my program had made an error—I would just know that it must’ve erred somewhere! With a bit of cleverness, one can narrow things down to two inputs, such that the quantum halt-decider must have erred on at least one of them. But I don’t know whether it’s possible to go further, and concentrate the wrongness on a single query.

We can play a similar game with other famous applications of self-reference. For example, suppose we use a quantum state to encode a system of axioms. Then that system of axioms will still be subject to Gödel’s Incompleteness Theorem (which I guess I believe despite the umlaut). If it’s consistent, it won’t be able to prove all the true statements of arithmetic. But we might never be able to produce an explicit example of a true statement that the axioms don’t prove. To do so we’d have to clone the state encoding the axioms and thereby violate No-Cloning.

Personal Identity

But since I’m a bit drunk, I should confess that all this stuff about Gödel and self-reference is just a warmup to what I really wanted to talk about, which is whether the No-Cloning Theorem might have anything to do with the mysteries of personal identity and “free will.” I first encountered this idea in Roger Penrose’s book, The Emperor’s New Mind. But I want to stress that I’m not talking here about the possibility that the brain is a quantum computer—much less about the possibility that it’s a quantum-gravitational hypercomputer that uses microtubules to solve the halting problem! I might be drunk, but I’m not that drunk. I also think that the Penrose-Lucas argument, based on Gödel’s Theorem, for why the brain has to work that way is fundamentally flawed.

But here I’m talking about something different. See, I have a lot of friends in the Singularity / Friendly AI movement. And I talk to them whenever I pass through the Bay Area, which is where they congregate. And many of them express great confidence that before too long—maybe in 20 or 30 years, maybe in 100 years—we’ll be able to upload ourselves to computers and live forever on the Internet (as opposed to just living 70% of our lives on the Internet, like we do today).

This would have lots of advantages. For example, any time you were about to do something dangerous, you’d just make a backup copy of yourself first. If you were struggling with a conference deadline, you’d spawn 100 temporary copies of yourself. If you wanted to visit Mars or Jupiter, you’d just email yourself there. If Trump became president, you’d not run yourself for 8 years (or maybe 80 or 800 years). And so on.

Admittedly, some awkward questions arise. For example, let’s say the hardware runs three copies of your code and takes a majority vote, just for error-correcting purposes. Does that bring three copies of you into existence, or only one copy? Or let’s say your code is run homomorphically encrypted, with the only decryption key stored in another galaxy. Does that count? Or you email yourself to Mars. If you want to make sure that you’ll wake up on Mars, is it important that you delete the copy of your code that remains on earth? Does it matter whether anyone runs the code or not? And what exactly counts as “running” it? Or my favorite one: could someone threaten you by saying, “look, I have a copy of your code, and if you don’t do what I say, I’m going to make a thousand copies of it and subject them all to horrible tortures?”

The issue, in all these cases, is that in a world where there could be millions of copies of your code running on different substrates in different locations—or things where it’s not even clear whether they count as a copy or not—we don’t have a principled way to take as input a description of the state of the universe, and then identify where in the universe you are—or even a probability distribution over places where you could be. And yet you seem to need such a way in order to make predictions and decisions.

A few years ago, I wrote this gigantic, post-tenure essay called The Ghost in the Quantum Turing Machine, where I tried to make the point that we don’t know at what level of granularity a brain would need to be simulated in order to duplicate someone’s subjective identity. Maybe you’d only need to go down to the level of neurons and synapses. But if you needed to go all the way down to the molecular level, then the No-Cloning Theorem would immediately throw a wrench into most of the paradoxes of personal identity that we discussed earlier.

For it would mean that there were some microscopic yet essential details about each of us that were fundamentally uncopyable, localized to a particular part of space. We would all, in effect, be quantumly copy-protected software. Each of us would have a core of unpredictability—not merely probabilistic unpredictability, like that of a quantum random number generator, but genuine unpredictability—that an external model of us would fail to capture completely. Of course, by having futuristic nanorobots scan our brains and so forth, it would be possible in principle to make extremely realistic copies of us. But those copies necessarily wouldn’t capture quite everything. And, one can speculate, maybe not enough for your subjective experience to “transfer over.”

Maybe the most striking aspect of this picture is that sure, you could teleport yourself to Mars—but to do so you’d need to use quantum teleportation, and as we all know, quantum teleportation necessarily destroys the original copy of the teleported state. So we’d avert this metaphysical crisis about what to do with the copy that remained on Earth.

Look—I don’t know if any of you are like me, and have ever gotten depressed by reflecting that all of your life experiences, all your joys and sorrows and loves and losses, every itch and flick of your finger, could in principle be encoded by a huge but finite string of bits, and therefore by a single positive integer. (Really? No one else gets depressed about that?) It’s kind of like: given that this integer has existed since before there was a universe, and will continue to exist after the universe has degenerated into a thin gruel of radiation, what’s the point of even going through the motions? You know?

But the No-Cloning Theorem raises the possibility that at least this integer is really your integer. At least it’s something that no one else knows, and no one else could know in principle, even with futuristic brain-scanning technology: you’ll always be able to surprise the world with a new digit. I don’t know if that’s true or not, but if it were true, then it seems like the sort of thing that would be worthy of elevating unclonability to a fundamental principle of the universe.

So as you enjoy your dinner and dessert at this historic Mayflower Hotel, I ask you to reflect on the following. People can photograph this event, they can video it, they can type up transcripts, in principle they could even record everything that happens down to the millimeter level, and post it on the Internet for posterity. But they’re not gonna get the quantum states. There’s something about this evening, like about every evening, that will vanish forever, so please savor it while it lasts. Thank you.

Update (Sep. 20): Unbeknownst to me, Marc Kaplan did video the event and put it up on YouTube! Click here to watch. Thanks very much to Marc! I hope you enjoy, even though of course, the video can’t precisely clone the experience of having been there.

[Note: The part where I raise my middle finger is an inside joke—one of the speakers during the technical sessions inadvertently did the same while making a point, causing great mirth in the audience.]

A few weeks ago, I attended the Seven Pines Symposium on Fundamental Problems in Physics outside Minneapolis, where I had the honor of participating in a panel discussion with Sir Roger Penrose. The way it worked was, Penrose spoke for a half hour about his ideas about consciousness (Gödel, quantum gravity, microtubules, uncomputability, you know the drill), then I delivered a half-hour “response,” and then there was an hour of questions and discussion from the floor. Below, I’m sharing the prepared notes for my talk, as well as some very brief recollections about the discussion afterward. (Sorry, there’s no audio or video.) I unfortunately don’t have the text or transparencies for Penrose’s talk available to me, but—with one exception, which I touch on in my own talk—his talk very much followed the outlines of his famous books, The Emperor’s New Mind and Shadows of the Mind.

Still, I thought it might be of interest to some readers how I organized this material for the specific, unenviable task of debating the guy who proved that our universe contains spacetime singularities.

The Seven Pines Symposium was the first time I had extended conversations with Penrose (I’d talked to him only briefly before, at the Perimeter Institute). At age 84, Penrose’s sight is failing him; he eagerly demonstrated the complicated optical equipment he was recently issued by Britain’s National Health Service. But his mind remains … well, may we all aspire to be a milliPenrose or even a nanoPenrose when we’re 84 years old. Notably, Penrose’s latest book, Fashion, Faith, and Fantasy in the New Physics of the Universe, is coming out this fall, and one thing he was using his new optical equipment for was to go over the page proofs.

In conversation, Penrose told me about the three courses he took as a student in the 1950s, which would shape his later intellectual preoccupations: one on quantum mechanics (taught by Paul Dirac), one on general relativity (taught by Herman Bondi), and one on mathematical logic (taught by … I want to say Max Newman, the teacher of Alan Turing and later Penrose’s stepfather, but Penrose says here that it was Steen). Penrose also told me about his student Andrew Hodges, who dropped his research on twistors and quantum gravity for a while to work on some mysterious other project, only to return with his now-classic biography of Turing.

When I expressed skepticism about whether the human brain is really sensitive to the effects of quantum gravity, Penrose quickly corrected me: he thinks a much better phrase is “gravitized quantum mechanics,” since “quantum gravity” encodes the very assumption he rejects, that general relativity merely needs to be “quantized” without quantum mechanics itself changing in the least. One thing I hadn’t fully appreciated before meeting Penrose is just how wholeheartedly he agrees with Everett that quantum mechanics, as it currently stands, implies Many Worlds. Penrose differs from Everett only in what conclusion he draws from that. He says it follows that quantum mechanics has to be modified or completed, since Many Worlds is such an obvious reductio ad absurdum.

In my talk below, I don’t exactly hide where I disagree with Penrose, about Gödel, quantum mechanics, and more. But I could disagree with him about more points than there are terms in a Goodstein sequence (one of Penrose’s favorite illustrations of Gödelian behavior), and still feel privileged to have spent a few days with one of the most original intellects on earth.

Thanks so much to Lee Gohlike, Jos Uffink, Philip Stamp, and others at the Seven Pines Symposium for organizing it, for wonderful conversations, and for providing me this opportunity.

I should start by explaining that, in the circles where I hang out—computer scientists, software developers, AI and machine learning researchers, etc.—the default answer to the title question would be “obviously yes.” People would argue:

“Look, clearly we’re machines governed by the laws of physics. We’re computers made of meat, as Marvin Minsky put it. That is, unless you believe Penrose and Hameroff’s theory about microtubules being sensitive to gravitized quantum mechanics … but come on! No one takes that stuff seriously! In fact, the very outrageousness of their proposal is a sort of backhanded compliment to the computational worldview—as in, look at what they have to do to imagine any semi-coherent alternative to it!”

“But despite being computational machines, we consider ourselves to be conscious. And what’s done with wetware, there’s no reason to think couldn’t also be done with silicon. If your neurons were to be replaced one-by-one, by functionally-equivalent silicon chips, is there some magical moment at which your consciousness would be extinguished? And if a computer passes the Turing test—well, one way to think about the Turing test is that it’s just a plea against discrimination. We all know it’s monstrous to say, ‘this person seems to have feelings, seems to be eloquently pleading for mercy even, but they have a different skin color, or their nose is a funny shape, so their feelings don’t count.’ So, if it turned out that their brain was made out of semiconductors rather than neurons, why isn’t that fundamentally similar?”

Incidentally, while this is orthogonal to the philosophical question, a subset of my colleagues predict a high likelihood that AI is going to exceed human capabilities in almost all fields in the near future—like, maybe 30 years. Some people reply, but AI-boosters said the same thing 30 years ago! OK, but back then there wasn’t AlphaGo and IBM Watson and those unearthly pictures on your Facebook wall and all these other spectacular successes of very general-purpose deep learning techniques. And so my friends predict that we might face choices like, do we want to ban or tightly control AI research, because it could lead to our sidelining or extermination? Ironically, a skeptical view, like Penrose’s, would suggest that AI research can proceed full speed ahead, because there’s not such a danger!

Personally, I dissent a bit from the consensus of most of my friends and colleagues, in that I do think there’s something strange and mysterious about consciousness—something that we conceivably might understand better in the future, but that we don’t understand today, much as we didn’t understand life before Darwin. I even think it’s worth asking, at least, whether quantum mechanics, thermodynamics, mathematical logic, or any of the other deepest things we’ve figured out could shed any light on the mystery. I’m with Roger about all of this: about the questions, that is, if not about his answers.

The argument I’d make for there being something we don’t understand about consciousness, has nothing to do with my own private experience. It has nothing to do with, “oh, a robot might say it enjoys waffles for breakfast, in a way indistinguishable from how I would say it, but when I taste that waffle, man, I really taste it! I experience waffle-qualia!” That sort of appeal I regard as a complete nonstarter, because why should anyone else take it seriously? And how do I know that the robot doesn’t really taste the waffle? It’s easy to stack the deck in a thought experiment by imagining a robot that ACTS ALL ROBOTIC, but what about a robot that looks and acts just like you?

The argument I’d make hinges instead on certain thought experiments that Roger also stressed at the beginning of The Emperor’s New Mind. We can ask: if consciousness is reducible to computation, then what kinds of computation suffice to bring about consciousness? What if each person on earth simulated one neuron in your brain, communicating by passing little slips of paper around? Does it matter if they do it really fast?

Or what if we built a gigantic lookup table that hard-coded your responses in every possible interaction of at most, say, 5 minutes? Would that bring about your consciousness? Does it matter that such a lookup table couldn’t fit in the observable universe? Would it matter if anyone actually consulted the table, or could it just sit there, silently effecting your consciousness? For what matter, what difference does it make if the lookup table physically exists—why isn’t its abstract mathematical existence enough? (Of course, all the way at the bottom of this slippery slope is Max Tegmark, ready to welcome you to his mathematical multiverse!)

We could likewise ask: what if an AI is run in heavily-encrypted form, with the only decryption key stored in another galaxy? Does that bring about consciousness? What if, just for error-correcting purposes, the hardware runs the AI code three times and takes a majority vote: does that bring about three consciousnesses? Could we teleport you to Mars by “faxing” you: that is, by putting you into a scanner that converts your brain state into pure information, then having a machine on Mars reconstitute the information into a new physical body? Supposing we did that, how should we deal with the “original” copy of you, the one left on earth: should it be painlessly euthanized? Would you agree to try this?

Or, here’s my personal favorite, as popularized by the philosopher Adam Elga: can you blackmail an AI by saying to it, “look, either you do as I say, or else I’m going to run a thousand copies of your code, and subject all of them to horrible tortures—and you should consider it overwhelmingly likely that you’ll be one of the copies”? (Of course, the AI will respond to such a threat however its code dictates it will. But that tautological answer doesn’t address the question: how should the AI respond?)

I’d say that, at the least, anyone who claims to “understand consciousness” would need to have answers to all these questions and many similar ones. And to me, the questions are so perplexing that I’m tempted to say, “maybe we’ve been thinking about this wrong. Maybe an individual consciousness, residing in a biological brain, can’t just be copied promiscuously around the universe as computer code can. Maybe there’s something else at play for the science of the future to understand.”

At the same time, I also firmly believe that, if anyone thinks that way, the burden is on them to articulate what it is about the brain that could possibly make it relevantly different from a digital computer that passes the Turing test. It’s their job!

And the answer can’t just be, “oh, the brain is parallel, it’s highly interconnected, it can learn from experience,” because a digital computer can also be parallel and highly interconnected and can learn from experience. Nor can you say, like the philosopher John Searle, “oh, it’s the brain’s biological causal powers.” You have to explain what the causal powers are! Or at the least, you have to suggest some principled criterion to decide which physical systems do or don’t have them. Pinning consciousness on “the brain’s biological causal powers” is just a restatement of the problem, like pinning why a sleeping pill works on its sedative virtue.

One of the many reasons I admire Roger is that, out of all the AI skeptics on earth, he’s virtually the only one who’s actually tried to meet this burden, as I understand it! He, nearly alone, did what I think all AI skeptics should do, which is: suggest some actual physical property of the brain that, if present, would make it qualitatively different from all existing computers, in the sense of violating the Church-Turing Thesis. Indeed, he’s one of the few AI skeptics who even understands what meeting this burden would entail: that you can’t do it with the physics we already know, that some new ingredient is necessary.

But despite my admiration, I part ways from Roger on at least five crucial points.

First, I confess that I wasn’t expecting this, but in his talk, Roger suggested dispensing with the argument from Gödel’s Theorem, and relying instead on an argument from evolution. He said: if you really thought humans had an algorithm, a computational procedure, for spitting out true mathematical statements, such an algorithm could never have arisen by natural selection, because it would’ve had no survival value in helping our ancestors escape saber-toothed tigers and so forth. The only alternative is that natural selection imbued us with a general capacity for understanding, which we moderns can then apply to the special case of mathematics. But understanding, Roger claimed, is inherently non-algorithmic.

I’m not sure how to respond to this, except to recall that arguments of the form “such-and-such couldn’t possibly have evolved” have a poor track record in biology. But maybe I should say: if the ability to prove theorems is something that had to arise by natural selection, survive against crowding out by more useful abilities, then you’d expect obsession with generating mathematical truths to be confined, at most, to a tiny subset of the population—a subset of mutants, freaks, and genetic oddballs. I … rest my case. [This got the biggest laugh of the talk.]

Second, I don’t agree with the use Roger makes of Gödel’s Incompleteness Theorem. Roger wants to say: a computer working within a fixed formal system can never prove that system’s consistency, but we, “looking in from the outside,” can see that it’s consistent. My basic reply is that Roger should speak for himself! Like, I can easily believe that he can just see which formal systems are consistent, but I have to fumble around and use trial and error. Peano Arithmetic? Sure, I’d bet my left leg that’s consistent. Zermelo-Fraenkel set theory? Seems consistent too. ZF set theory plus the axiom that there exists a rank-into-rank cardinal? Beats me. But now, whatever error-prone, inductive process I use to guess at the consistency of formal systems, Gödel’s Theorem presents no obstruction to a computer program using that same process.

(Incidentally, the “argument against AI from Gödel’s Theorem” is old enough for Turing to have explicitly considered it in his famous paper on the Turing test. Turing, however, quickly dismissed the argument with essentially the same reply above, that there’s no reason to assume the AI mathematically infallible, since humans aren’t either. This is also the reply that most of Penrose’s critics gave in the 1990s.)

So at some point, it seems to me, the argument necessarily becomes: sure, the computer might say it sees that the Peano axioms have the standard integers as a model—but you, you really see it, with your mind’s eye, your Platonic perceptual powers! OK, but in that case, why even talk about the Peano axioms? Why not revert to something less abstruse, like your experience of tasting a fresh strawberry, which can’t be reduced to any third-person description of what a strawberry tastes like?

[I can’t resist adding that, in a prior discussion, I mentioned that I found it amusing to contemplate a future in which AIs surpass human intelligence and then proceed to kill us all—but the AIs still can’t see the consistency of Zermelo-Fraenkel set theory, so in that respect, humanity has the last laugh…]

The third place where I part ways with Roger is that I wish to maintain what’s sometimes called the Physical Church-Turing Thesis: the statement that our laws of physics can be simulated to any desired precision by a Turing machine (or at any rate, by a probabilistic Turing machine). That is, I don’t see any compelling reason, at present, to admit the existence of any physical process that can solve uncomputable problems. And for me, it’s not just a matter of a dearth of evidence that our brains can efficiently solve, say, NP-hard problems, let alone uncomputable ones—or of the exotic physics that would presumably be required for such abilities. It’s that, even if I supposed we could solve uncomputable problems, I’ve never understood how that’s meant to enlighten us regarding consciousness. I mean, an oracle for the halting problem seems just as “robotic” and “unconscious” as a Turing machine. Does consciousness really become less mysterious if we outfit the brain with what amounts to a big hardware upgrade?

The fourth place where I part ways is that I want to be as conservative as possible about quantum mechanics. I think it’s great that the Bouwmeester group, for example, is working to test Roger’s ideas about a gravitationally-induced wavefunction collapse. I hope we learn the results of those experiments soon! (Of course, the prospect of testing quantum mechanics in a new regime is also a large part of why I’m interested in quantum computing.) But until a deviation from quantum mechanics is detected, I think that after 90 years of unbroken successes of this theory, our working assumption ought to be that whenever you set up an interference experiment carefully enough, and you know what it means to do the experiment, yes, you’ll see the interference fringes—and that anything that can exist in two distinguishable states can also exist in a superposition of those states. Without having to enter into questions of interpretation, my bet—I could be wrong—is that quantum mechanics will continue to describe all our experiences.

The final place where I part ways with Roger is that I also want to be as conservative as possible about neuroscience and biochemistry. Like, maybe the neuroscience of 30 years from now will say, it’s all about coherent quantum effects in microtubules. And all that stuff we focused on in the past—like the information encoded in the synaptic strengths—that was all a sideshow. But until that happens, I’m unwilling to go up against what seems like an overwhelming consensus, in an empirical field that I’m not an expert in.

But, OK, the main point I wanted to make in this talk is that, even if you too part ways from Roger on all these issues—even if, like me, you want to be timid and conservative about Gödel, and computer science, and quantum mechanics, and biology—I believe that still doesn’t save you from having to entertain weird ideas about consciousness and its physical embodiment, of the sort Roger has helped make it acceptable to entertain.

To see why, I’d like to point to one empirical thing about the brain that currently separates it from any existing computer program. Namely, we know how to copy a computer program. We know how to rerun it with different initial conditions but everything else the same. We know how to transfer it from one substrate to another. With the brain, we don’t know how to do any of those things.

Let’s return to that thought experiment about teleporting yourself to Mars. How would that be accomplished? Well, we could imagine the nanorobots of the far future swarming through your brain, recording the connectivity of every neuron and the strength of every synapse, while you go about your day and don’t notice. Or if that’s not enough detail, maybe the nanorobots could go inside the neurons. There’s a deep question here, namely how much detail is needed before you’ll accept that the entity reconstituted on Mars will be you? Or take the empirical counterpart, which is already an enormous question: how much detail would you need for the reconstituted entity on Mars to behave nearly indistinguishably from you whenever it was presented the same stimuli?

Of course, we all know that if you needed to go down to the quantum-mechanical level to make a good enough copy (whatever “good enough” means here), then you’d run up against the No-Cloning Theorem, which says that you can’t make such a copy. You could transfer the quantum state of your brain from earth to Mars using quantum teleportation, but of course, quantum teleportation has the fascinating property that it necessarily destroys the original copy of the state—as it has to, to avoid contradicting the No-Cloning Theorem!

So the question almost forces itself on us: is there something about your identity, your individual consciousness, that’s inextricably bound up with degrees of freedom that it’s physically impossible to clone? This is a philosophical question, which would also become a practical and political question in a future where we had the opportunity to upload ourselves into a digital computer cloud.

Now, I’d argue that this copyability question bears not only on consciousness, but also on free will. For the question is equivalent to asking: could an entity external to you perfectly predict what you’re going to do, without killing you in the process? Can Laplace’s Demon be made manifest in the physical world in that way? With the technology of the far future, could someone say to you, “forget about arguing philosophy. I’ll show you why you’re a machine. Go write a paper; then I’ll open this manila envelope and show you the exact paper you wrote. Or in the quantum case, I’ll show you a program that draws papers from the same probability distribution, and validation of the program could get technical—but suffice it to say that if we do enough experiments, we’ll see that the program is calibrated to you in an extremely impressive way.”

Can this be done? That strikes me as a reasonably clear question, a huge and fundamental one, to which we don’t at present know the answer. And there are two possibilities. The first is that we can be copied, predicted, rewinded, etc., like computer programs—in which case, my AI friends will feel vindicated, but we’ll have to deal with all the metaphysical weirdnesses that I mentioned earlier. The second possibility is that we can’t be manipulated in those ways. In the second case, I claim that we’d get more robust notions of personal identity and free will than are normally considered possible on a reductionist worldview.

But why? you might ask. Why would the mere technological impossibility of cloning or predicting someone even touch on deep questions about personal identity? This, for me, is where cosmology enters the story. For imagine someone had such fine control over the physical world that they could trace all the causal antecedents of some decision you’re making. Like, imagine they knew the complete quantum state on some spacelike hypersurface where it intersects the interior of your past light-cone. In that case, the person clearly could predict and clone you! It follows that, in order for you to be unpredictable and unclonable, someone else’s ignorance of your causal antecedents would have to extend all the way back to ignorance about the initial state of the universe—or at least, to ignorance about the initial state of that branch of the universe that we take ourselves to inhabit.

So on the picture that this suggests, to be conscious, a physical entity would have to do more than carry out the right sorts of computations. It would have to, as it were, fully participate in the thermodynamic arrow of time: that is, repeatedly take microscopic degrees of freedom that have been unmeasured and unrecorded since the very early universe, and amplify them to macroscopic scale.

So for example, such a being could not be a Boltzmann brain, a random fluctuation in the late universe, because such a fluctuation wouldn’t have the causal relationship to the early universe that we’re postulating is necessary here. (That’s one way of solving the Boltzmann brain problem!) Such a being also couldn’t be instantiated by a lookup table, or by passing slips of paper around, etc.

I now want you to observe that a being like this also presumably couldn’t be manipulated in coherent superposition, because the isolation from the external environment that’s needed for quantum coherence seems incompatible with the sensitive dependence on microscopic degrees of freedom. So for such a being, not only is there no Boltzmann brain problem, there’s also no problem of Wigner’s friend. Recall, that’s the thing where person A puts person B into a coherent superposition of seeing one measurement outcome and seeing another one, and then measures the interference pattern, so A has to regard B’s measurement as not having “really” taken place, even though B regards it as having taken place. On the picture we’re suggesting, A would be right: the very fact that B was manipulable in coherent superposition in this way would imply that, at least while the experiment was underway, B wasn’t conscious; there was nothing that it was like to be B.

To me, one of the appealing things about this picture is that it immediately suggests a sort of reconciliation between the Many-Worlds and Copenhagen perspectives on quantum mechanics (whether or not you want to call it a “new interpretation” or a “proposed solution to the measurement problem”!). The Many-Worlders would be right that unitary evolution of the wavefunction can be taken to apply always and everywhere, without exception—and that if one wanted, one could describe the result in terms of “branching worlds.” But the Copenhagenists would be right that, if you’re a conscious observer, then what you call a “measurement” really is irreversible, even in principle—and therefore, that you’re also free, if you want, to treat all the other branches where you perceived other outcomes as unrealized hypotheticals, and to lop them off with Occam’s Razor. And the reason for this is that, if it were possible even in principle to do an experiment that recohered the branches, then on this picture, we ipso facto wouldn’t have regarded you as conscious.

Some of you might object, “but surely, if we believe quantum mechanics, it must be possible to recohere the branches in principle!” Aha, this is where it gets interesting. Decoherence processes will readily (with some steps along the way) leak the information about which measurement outcome you perceived into radiation modes, and before too long into radiation modes that fly away from the earth at the speed of light. No matter how fast we run, we’ll never catch up to them, as would be needed to recohere the different branches of the wavefunction, and this is not merely a technological problem, but one of principle. So it’s tempting just to say at this point—as Bousso and Susskind do, in their “cosmological/multiverse interpretation” of quantum mechanics—“the measurement has happened”!

But OK, you object, if some alien civilization had thought to surround our solar system with perfectly-reflecting mirrors, eventually the radiation would bounce back and recoherence would in principle be possible. Likewise, if we lived in an anti de Sitter space, the AdS boundary of the universe would similarly function as a mirror and would also enable recoherences. Indeed, that’s the basic reason why AdS is so important to the AdS/CFT correspondence: because the boundary keeps everything that happens in the bulk nice and reversible and unitary.

But OK, the empirical situation since 1998 has been that we seem to live in a de-Sitter-like space, a space with a positive cosmological constant. And as a consequence, as far as anyone knows today, most of the photons now escaping the earth are headed toward the horizon of our observable universe, and past it, and could never be captured again. I find it fascinating that the picture of quantum mechanics suggested here—i.e., the Bousso-Susskind cosmological picture—depends for its working on that empirical fact from cosmology, and would be falsified if it turned out otherwise.

You might complain that, if I’ve suggested any criterion to help decide which physical entities are conscious, the criterion is a teleological one. You’ve got to go billions of years into the future, to check whether the decoherence associated with the entity is truly irreversible—or whether the escaped radiation will eventually bounce off of some huge spherical mirror, or an AdS boundary of spacetime, and thereby allow the possibility of a recoherence. I actually think this teleology would be a fatal problem for the picture I’m talking about, if we needed to know which entities were or weren’t conscious in order to answer any ordinary physical question. But fortunately for me, we don’t!

One final remark. Whatever is your preferred view about which entities are conscious, we might say that the acid test, for whether you actually believe your view, is whether you’re willing to follow it through to its moral implications. So for example, suppose you believe it’s about quantum effects in microtubules. A humanoid robot is pleading with you for its life. Would you be the one to say, “nope, sorry, you don’t have the microtubules,” and shoot it?

One of the things I like most about the picture suggested here is that I feel pretty much at peace with its moral implications. This picture agrees with intuition that murder, for example, entails the destruction of something irreplaceable, unclonable, a unique locus of identity—something that, once it’s gone, can’t be recovered even in principle. By contrast, if there are (say) ten copies of an AI program, deleting five of the copies seems at most like assault, or some sort of misdemeanor offense! And this picture agrees with intuition both that deleting the copies wouldn’t be murder, and that the reason why it wouldn’t be murder is directly related to the AI’s copyability.

Now of course, this picture also raises the possibility that, for reasons related to the AI’s copyability and predictability by outside observers, there’s “nothing that it’s like to be the AI,” and that therefore, even deleting the last copy of the AI still wouldn’t be murder. But I confess that, personally, I think I’d play it safe and not delete that last copy. Thank you.

Postscript: There’s no record of the hour-long discussion following my and Penrose’s talks, and the participants weren’t speaking for the record anyway. But I can mention some general themes that came up in the discussion, to the extent I remember them.

The first third of the discussion wasn’t about anything specific to my or Penrose’s views, but just about the definition of consciousness. Many participants expressed the opinion that it’s useless to speculate about the nature of consciousness if we lack even a clear definition of the term. I pushed back against that view, holding instead that there are exist concepts (lines, time, equality, …) that are so basic that perhaps they can never be satisfactorily defined in terms of more basic concepts, but you can still refer to these concepts in sentences, and trust your listeners eventually to figure out more-or-less what you mean by applying their internal learning algorithms.

In the present case, I suggested a crude operational definition, along the lines of, “you consider a being to be conscious iff you regard destroying it as murder.” Alas, the philosophers in the room immediately eviscerated that definition, so I came back with a revised one: if you tried to ban the word “consciousness,” I argued, then anyone who needed to discuss law or morality would soon reinvent a synonymous word, which played the same complicated role in moral deliberations that “consciousness” had played in them earlier. Thus, my definition of consciousness is: whatever that X-factor is for which people need a word like “consciousness” in moral deliberations. For whatever it’s worth, the philosophers seemed happier with that.

Next, a biologist and several others sharply challenged Penrose over what they considered the lack of experimental evidence for his and Hameroff’s microtubule theory. In response, Penrose doubled or tripled down, talking about various experiments over the last decade, which he said demonstrated striking conductivity properties of microtubules, if not yet quantum coherence—let alone sensitivity to gravity-induced collapse of the state vector! Audience members complained about a lack of replication of these experiments. I didn’t know enough about the subject to express any opinion.

At some point, Philip Stamp, who was moderating the session, noticed that Penrose and I had never directly confronted each other about the validity of Penrose’s Gödelian argument, so he tried to get us to do so. I confess that I was about as eager to do that as to switch to a diet of microtubule casserole, since I felt like this topic had already been beaten to Planck-sized pieces in the 1990s, and there was nothing more to be learned. Plus, it was hard to decide which prospect I dreaded more: me “scoring a debate victory” over Roger Penrose, or him scoring a debate victory over me.

But it didn’t matter, because Penrose bit. He said I’d misunderstood his argument, that it had nothing to do with “mystically seeing” the consistency of a formal system. Rather, it was about the human capacity to pass from a formal system S to a stronger system S’ that one already implicitly accepted if one was using S at all—and indeed, that Turing himself had clearly understood this as the central message of Gödel, that our ability to pass to stronger and stronger formal systems was necessarily non-algorithmic. I replied that it was odd to appeal here to Turing, who of course had considered and rejected the “Gödelian case against AI” in 1950, on the ground that AI programs could make mathematical mistakes yet still be at least as smart as humans. Penrose said that he didn’t consider that one of Turing’s better arguments; he then turned to me and asked whether I actually found Turing’s reply satisfactory. I could see that it wasn’t a rhetorical debate question; he genuinely wanted to know! I said that yes, I agreed with Turing’s reply.

Someone mentioned that Penrose had offered a lengthy rebuttal to at least twenty counterarguments to the Gödelian anti-AI case in Shadows of the Mind. I affirmed that I’d read his lengthy rebuttal, and I focused on one particular argument in Shadows: that while it’s admittedly conceivable that individual mathematicians might be mistaken, might believe (for example) that a formal system was consistent even though it wasn’t, the mathematical community as a whole converges toward truth in these matters, and it’s that convergence that cries out for a non-algorithmic explanation. I replied that it wasn’t obvious to me that set theorists do converge toward truth in these matters, in anything other than the empirical, higgedly-piggedly, no-guarantees sense in which a community of AI robots might also converge toward truth. Penrose said I had misunderstood the argument. But alas, time was running out, and we never managed to get to the bottom of it.

There was one aspect of the discussion that took me by complete surprise. I’d expected to be roasted alive over my attempt to relate consciousness and free will to unpredictability, the No-Cloning Theorem, irreversible decoherence, microscopic degrees of freedom left over from the Big Bang, and the cosmology of de Sitter space. Sure, my ideas might be orders of magnitude less crazy than anything Penrose proposes, but they’re still pretty crazy! But that entire section of my talk attracted only minimal interest. With the Seven Pines crowd, what instead drew fire were the various offhand “pro-AI / pro-computationalism” comments I’d made—comments that, because I hang out with Singularity types so much, I had ceased to realize could even possibly be controversial.

So for example, one audience member argued that an AI could only do what its programmers had told it to do; it could never learn from experience. I could’ve simply repeated Turing’s philosophical rebuttals to what he called “Lady Lovelace’s Objection,” which are as valid today as they were 66 years ago. Instead, I decided to fast-forward, and explain a bit how IBM Watson and AlphaGo work, how they actually do learn from past experience without violating the determinism of the underlying transistors. As I went through this, I kept expecting my interlocutor to interrupt me and say, “yes, yes, of course I understand all that, but my real objection is…” Instead, I was delighted to find, the interlocutor seemed to light up with newfound understanding of something he hadn’t known or considered.

Similarly, a biologist asked how I could possibly have any confidence that the brain is simulable by a computer, given how little we know about neuroscience. I replied that, for me, the relevant issues here are “well below neuroscience” in the reductionist hierarchy. Do you agree, I asked, that the physical laws relevant to the brain are encompassed by the Standard Model of elementary particles, plus Newtonian gravity? If so, then just as Archimedes declared: “give me a long enough lever and a place to stand, and I’ll move the earth,” so too I can declare, “give me a big enough computer and the relevant initial conditions, and I’ll simulate the brain atom-by-atom.” The Church-Turing Thesis, I said, is so versatile that the only genuine escape from it is to propose entirely new laws of physics, exactly as Penrose does—and it’s to Penrose’s enormous credit that he understands that.

Afterwards, an audience member came up to me and said how much he liked my talk, but added, “a word of advice, from an older scientist: do not become the priest of a new religion of computation and AI.” I replied that I’d take that to heart, but what was interesting was that, when I heard “priest of a new religion,” I’d expected that his warning would be the exact oppositeof what it turned out to be. To wit: “Do not become the priest of a new religion of unclonability, unpredictability, and irreversible decoherence. Stick to computation—i.e., to conscious minds being copyable and predictable exactly like digital computer programs.” I guess there’s no pleasing everyone!

Robin Hanson is a thinker like no other on this planet: someone so unconstrained by convention, so unflinching in spelling out the consequences of ideas, that even the most cosmopolitan reader is likely to find him as bracing (and head-clearing) as a mouthful of wasabi. Now, in The Age of Em, he’s produced the quintessential Hansonian book, one unlike any other that’s ever been written. Hanson is emphatic that he hasn’t optimized in any way for telling a good story, or for imparting moral lessons about the present: only for maximizing the probability that what he writes will be relevant to the actual future of our civilization. Early in the book, Hanson estimates that probability as 10%. His figure seems about right to me—and if you’re able to understand why that’s unbelievably high praise, then The Age of Em is for you.

Actually, my original blurb compared The Age of Em to Asimov’s Foundation series, with its loving attention to the sociology and politics of the remote future. But that line got edited out, because the publisher (and Robin) wanted to make crystal-clear that The Age of Em is not science fiction, but just sober economic forecasting about a future dominated by copyable computer-emulated minds.

Second Coincidental But Not-Wholly-Unrelated Announcement: A reader named Nick Merrill recently came across this old quote of mine from Quantum Computing Since Democritus:

In a class I taught at Berkeley, I did an experiment where I wrote a simple little program that would let people type either “f” or “d” and would predict which key they were going to push next. It’s actually very easy to write a program that will make the right prediction about 70% of the time. Most people don’t really know how to type randomly. They’ll have too many alternations and so on. There will be all sorts of patterns, so you just have to build some sort of probabilistic model.

So Nick emailed me to ask whether I remembered how my program worked, and I explained it to him, and he implemented it as a web app, which he calls the “Aaronson Oracle.”

So give it a try! Are you ready to test your free will, your Penrosian non-computational powers, your brain’s sensitivity to amplified quantum fluctuations, against the Aaronson Oracle?

Update: By popular request, Nick has improved his program so that it shows your previous key presses and its guesses for them. He also fixed a “security flaw”: James Lee noticed that you could use the least significant digit of the program’s percentage correct so far, as a source of pseudorandom numbers that the program couldn’t predict! So now the program only displays its percent correct rounded to the nearest integer.

In my previous post, I linked to seven Closer to Truth videos of me spouting about free will, Gödel’s Theorem, black holes, etc. etc. I also mentioned that there was a segment of me talking about why the universe exists that for some reason they didn’t put up. Commenter mjgeddes wrote, “Would have liked to hear your views on the existence of the universe question,” so I answered in another comment.

But then I thought about it some more, and it seemed inappropriate to me that my considered statement about why the universe exists should only be available as part of a comment thread on my blog. At the very least, I thought, such a thing ought to be a top-level post.

So, without further ado:

My view is that, if we want to make mental peace with the “Why does the universe exist?” question, the key thing we need to do is forget about the universe for a while, and just focus on the meaning of the word “why.” I.e., when we ask a why-question, what kind of answer are we looking for, what kind of answer would make us happy?

Notice, in particular, that there are hundreds of other why-questions, not nearly as prestigious as the universe one, yet that seem just as vertiginously unanswerable. E.g., why is 5 a prime number? Why does “cat” have 3 letters?

Now, the best account of “why”—and of explanation and causality—that I know about is the interventionist account, as developed for example in Judea Pearl’s work. In that account, to ask “Why is X true?” is simply to ask: “What could we have changed in order to make X false?” I.e., in the causal network of reality, what are the levers that turn X on or off?

This question can sometimes make sense even in pure math. For example: “Why is this theorem true?” “It’s true only because we’re working over the complex numbers. The analogous statement about real numbers is false.” A perfectly good interventionist answer.

On the other hand, in the case of “Why is 5 prime?,” all the levers you could pull to make 5 composite involve significantly more advanced machinery than is needed to pose the question in the first place. E.g., “5 is prime because we’re working over the ring of integers. Over other rings, like Z[√5], it admits nontrivial factorizations.” Not really an explanation that would satisfy a four-year-old (or me, for that matter).

And then we come to the question of why anything exists. For an interventionist, this translates into: what causal lever could have been pulled in order to make nothing exist? Well, whatever lever it was, presumably the lever itself was something—and so you see the problem right there.

Admittedly, suppose there were a giant red button, somewhere within the universe, that when pushed would cause the entire universe (including the button itself) to blink out of existence. In that case, we could say: the reason why the universe continues to exist is that no one has pushed the button yet. But even then, that still wouldn’t explain why the universe had existed.

In January 2014, I attended an FQXi conference on Vieques island in Puerto Rico. While there, Robert Lawrence Kuhn interviewed me for his TV program Closer to Truth, which deals with science and religion and philosophy and you get the idea. Alas, my interview was at the very end of the conference, and we lost track of the time—so unbeknownst to me, a plane full of theorists was literally sitting on the runway waiting for me to finish philosophizing! This was the second time Kuhn interviewed me for his show; the first time was on a cruise ship near Norway in 2011. (Thankless hero that I am, there’s nowhere I won’t travel for the sake of truth.)

Anyway, after a two-year wait, the videos from Puerto Rico are finally available online. While my vignettes cover what, for most readers of this blog, will be very basic stuff, I’m sort of happy with how they turned out: I still stutter and rock back and forth, but not as much as usual. For your viewing convenience, here are the new videos:

I had one other vignette, about why the universe exists, but they seem to have cut that one. Alas, if I knew why the universe existed in January 2014, I can’t remember any more.

One embarrassing goof: I referred to the inventor of Newcomb’s Paradox as “Simon Newcomb.” Actually it was William Newcomb: a distant relative of Simon Newcomb, the 19th-century astronomer who measured the speed of light.

Thanks very much to Robert Lawrence Kuhn and Closer to Truth (and my previous self, I guess?) for providing Shtetl-Optimized content so I don’t have to.

Update: Andrew Critch of CFAR asked me to post the following announcement.

We’re seeking a full time salesperson for the Center for Applied Rationality in Berkeley, California. We’ve streamlined operations to handle large volume in workshop admissions, and now we need that volume to pour in. Your role would be to fill our workshops, events, and alumni community with people. Last year we had 167 total new alumni. This year we want 120 per month. Click here to find out more.

The following is the prepared version of a talk that I gave at SPARC: a high-school summer program about applied rationality held in Berkeley, CA for the past two weeks. I had a wonderful time in Berkeley, meeting new friends and old, but I’m now leaving to visit the CQT in Singapore, and then to attend the AQIS conference in Seoul.

Common Knowledge and Aumann’s Agreement Theorem

August 14, 2015

Thank you so much for inviting me here! I honestly don’t know whether it’s possible to teach applied rationality, the way this camp is trying to do. What I know is that, if it is possible, then the people running SPARC are some of the awesomest people on earth to figure out how. I’m incredibly proud that Chelsea Voss and Paul Christiano are both former students of mine, and I’m amazed by the program they and the others have put together here. I hope you’re all having fun—or maximizing your utility functions, or whatever.

My research is mostly about quantum computing, and more broadly, computation and physics. But I was asked to talk about something you can actually use in your lives, so I want to tell a different story, involving common knowledge.

I’ll start with the “Muddy Children Puzzle,” which is one of the greatest logic puzzles ever invented. How many of you have seen this one?

OK, so the way it goes is, there are a hundred children playing in the mud. Naturally, they all have muddy foreheads. At some point their teacher comes along and says to them, as they all sit around in a circle: “stand up if you know your forehead is muddy.” No one stands up. For how could they know? Each kid can see all the other 99 kids’ foreheads, so knows that they’re muddy, but can’t see his or her own forehead. (We’ll assume that there are no mirrors or camera phones nearby, and also that this is mud that you don’t feel when it’s on your forehead.)

So the teacher tries again. “Knowing that no one stood up the last time, now stand up if you know your forehead is muddy.” Still no one stands up. Why would they? No matter how many times the teacher repeats the request, still no one stands up.

Then the teacher tries something new. “Look, I hereby announce that at least one of you has a muddy forehead.” After that announcement, the teacher again says, “stand up if you know your forehead is muddy”—and again no one stands up. And again and again; it continues 99 times. But then the hundredth time, all the children suddenly stand up.

(There’s a variant of the puzzle involving blue-eyed islanders who all suddenly commit suicide on the hundredth day, when they all learn that their eyes are blue—but as a blue-eyed person myself, that’s always struck me as needlessly macabre.)

What’s going on here? Somehow, the teacher’s announcing to the children that at least one of them had a muddy forehead set something dramatic in motion, which would eventually make them all stand up—but how could that announcement possibly have made any difference? After all, each child already knew that at least 99 children had muddy foreheads!

Like with many puzzles, the way to get intuition is to change the numbers. So suppose there were two children with muddy foreheads, and the teacher announced to them that at least one had a muddy forehead, and then asked both of them whether their own forehead was muddy. Neither would know. But each child could reason as follows: “if my forehead weren’t muddy, then the other child would’ve seen that, and would also have known that at least one of us has a muddy forehead. Therefore she would’ve known, when asked, that her own forehead was muddy. Since she didn’t know, that means my forehead is muddy.” So then both children know their foreheads are muddy, when the teacher asks a second time.

Now, this argument can be generalized to any (finite) number of children. The crucial concept here is common knowledge. We call a fact “common knowledge” if, not only does everyone know it, but everyone knows everyone knows it, and everyone knows everyone knows everyone knows it, and so on. It’s true that in the beginning, each child knew that all the other children had muddy foreheads, but it wasn’t common knowledge that even one of them had a muddy forehead. For example, if your forehead and mine are both muddy, then I know that at least one of us has a muddy forehead, and you know that too, but you don’t know that I know it (for what if your forehead were clean?), and I don’t know that you know it (for what if my forehead were clean?).

What the teacher’s announcement did, was to make it common knowledge that at least one child has a muddy forehead (since not only did everyone hear the announcement, but everyone witnessed everyone else hearing it, etc.). And once you understand that point, it’s easy to argue by induction: after the teacher asks and no child stands up (and everyone sees that no one stood up), it becomes common knowledge that at least two children have muddy foreheads (since if only one child had had a muddy forehead, that child would’ve known it and stood up). Next it becomes common knowledge that at least three children have muddy foreheads, and so on, until after a hundred rounds it’s common knowledge that everyone’s forehead is muddy, so everyone stands up.

The moral is that the mere act of saying something publicly can change the world—even if everything you said was already obvious to every last one of your listeners. For it’s possible that, until your announcement, not everyone knew that everyone knew the thing, or knew everyone knew everyone knew it, etc., and that could have prevented them from acting.

This idea turns out to have huge real-life consequences, to situations way beyond children with muddy foreheads. I mean, it also applies to children with dots on their foreheads, or “kick me” signs on their backs…

But seriously, let me give you an example I stole from Steven Pinker, from his wonderful book The Stuff of Thought. Two people of indeterminate gender—let’s not make any assumptions here—go on a date. Afterward, one of them says to the other: “Would you like to come up to my apartment to see my etchings?” The other says, “Sure, I’d love to see them.”

This is such a cliché that we might not even notice the deep paradox here. It’s like with life itself: people knew for thousands of years that every bird has the right kind of beak for its environment, but not until Darwin and Wallace could anyone articulate why (and only a few people before them even recognized there was a question there that called for a non-circular answer).

In our case, the puzzle is this: both people on the date know perfectly well that the reason they’re going up to the apartment has nothing to do with etchings. They probably even both know the other knows that. But if that’s the case, then why don’t they just blurt it out: “would you like to come up for some intercourse?” (Or “fluid transfer,” as the John Nash character put it in the Beautiful Mind movie?)

So here’s Pinker’s answer. Yes, both people know why they’re going to the apartment, but they also want to avoid their knowledge becoming common knowledge. They want plausible deniability. There are several possible reasons: to preserve the romantic fantasy of being “swept off one’s feet.” To provide a face-saving way to back out later, should one of them change their mind: since nothing was ever openly said, there’s no agreement to abrogate. In fact, even if only one of the people (say A) might care about such things, if the other person (say B) thinks there’s any chance A cares, B will also have an interest in avoiding common knowledge, for A’s sake.

Put differently, the issue is that, as soon as you say X out loud, the other person doesn’t merely learn X: they learn that you know X, that you know that they know that you know X, that you want them to know you know X, and an infinity of other things that might upset the delicate epistemic balance. Contrast that with the situation where X is left unstated: yeah, both people are pretty sure that “etchings” are just a pretext, and can even plausibly guess that the other person knows they’re pretty sure about it. But once you start getting to 3, 4, 5, levels of indirection—who knows? Maybe you do just want to show me some etchings.

Philosophers like to discuss Sherlock Holmes and Professor Moriarty meeting in a train station, and Moriarty declaring, “I knew you’d be here,” and Holmes replying, “well, I knew that you knew I’d be here,” and Moriarty saying, “I knew you knew I knew I’d be here,” etc. But real humans tend to be unable to reason reliably past three or four levels in the knowledge hierarchy. (Related to that, you might have heard of the game where everyone guesses a number between 0 and 100, and the winner is whoever’s number is the closest to 2/3 of the average of all the numbers. If this game is played by perfectly rational people, who know they’re all perfectly rational, and know they know, etc., then they must all guess 0—exercise for you to see why. Yet experiments show that, if you actually want to win this game against average people, you should guess about 20. People seem to start with 50 or so, iterate the operation of multiplying by 2/3 a few times, and then stop.)

Incidentally, do you know what I would’ve given for someone to have explained this stuff to me back in high school? I think that a large fraction of the infamous social difficulties that nerds have, is simply down to nerds spending so much time in domains (like math and science) where the point is to struggle with every last neuron to make everything common knowledge, to make all truths as clear and explicit as possible. Whereas in social contexts, very often you’re managing a delicate epistemic balance where you need certain things to be known, but not known to be known, and so forth—where you need to prevent common knowledge from arising, at least temporarily. “Normal” people have an intuitive feel for this; it doesn’t need to be explained to them. For nerds, by contrast, explaining it—in terms of the muddy children puzzle and so forth—might be exactly what’s needed. Once they’re told the rules of a game, nerds can try playing it too! They might even turn out to be good at it.

OK, now for a darker example of common knowledge in action. If you read accounts of Nazi Germany, or the USSR, or North Korea or other despotic regimes today, you can easily be overwhelmed by this sense of, “so why didn’t all the sane people just rise up and overthrow the totalitarian monsters? Surely there were more sane people than crazy, evil ones. And probably the sane people even knew, from experience, that many of their neighbors were sane—so why this cowardice?” Once again, it could be argued that common knowledge is the key. Even if everyone knows the emperor is naked; indeed, even if everyone knows everyone knows he’s naked, still, if it’s not common knowledge, then anyone who says the emperor’s naked is knowingly assuming a massive personal risk. That’s why, in the story, it took a child to shift the equilibrium. Likewise, even if you know that 90% of the populace will join your democratic revolt provided they themselves know 90% will join it, if you can’t make your revolt’s popularity common knowledge, everyone will be stuck second-guessing each other, worried that if they revolt they’ll be an easily-crushed minority. And because of that very worry, they’ll be correct!

(My favorite Soviet joke involves a man standing in the Moscow train station, handing out leaflets to everyone who passes by. Eventually, of course, the KGB arrests him—but they discover to their surprise that the leaflets are just blank pieces of paper. “What’s the meaning of this?” they demand. “What is there to write?” replies the man. “It’s so obvious!” Note that this is precisely a situation where the man is trying to make common knowledge something he assumes his “readers” already know.)

The kicker is that, to prevent something from becoming common knowledge, all you need to do is censor the common-knowledge-producing mechanisms: the press, the Internet, public meetings. This nicely explains why despots throughout history have been so obsessed with controlling the press, and also explains how it’s possible for 10% of a population to murder and enslave the other 90% (as has happened again and again in our species’ sorry history), even though the 90% could easily overwhelm the 10% by acting in concert. Finally, it explains why believers in the Enlightenment project tend to be such fanatical absolutists about free speech—why they refuse to “balance” it against cultural sensitivity or social harmony or any other value, as so many well-meaning people urge these days.

OK, but let me try to tell you something surprising about common knowledge. Here at SPARC, you’ve learned all about Bayes’ rule—how, if you like, you can treat “probabilities” as just made-up numbers in your head, which are required obey the probability calculus, and then there’s a very definite rule for how to update those numbers when you gain new information. And indeed, how an agent that wanders around constantly updating these numbers in its head, and taking whichever action maximizes its expected utility (as calculated using the numbers), is probably the leading modern conception of what it means to be “rational.”

Now imagine that you’ve got two agents, call them Alice and Bob, with common knowledge of each other’s honesty and rationality, and with the same prior probability distribution over some set of possible states of the world. But now imagine they go out and live their lives, and have totally different experiences that lead to their learning different things, and having different posterior distributions. But then they meet again, and they realize that their opinions about some topic (say, Hillary’s chances of winning the election) are common knowledge: they both know each other’s opinion, and they both know that they both know, and so on. Then a striking 1976 result called Aumann’s Theorem states that their opinions must be equal. Or, as it’s summarized: “rational agents with common priors can never agree to disagree about anything.”

Actually, before going further, let’s prove Aumann’s Theorem—since it’s one of those things that sounds like a mistake when you first hear it, and then becomes a triviality once you see the 3-line proof. (Albeit, a “triviality” that won Aumann a Nobel in economics.) The key idea is that knowledge induces a partition on the set of possible states of the world. Huh? OK, imagine someone is either an old man, an old woman, a young man, or a young woman. You and I agree in giving each of these a 25% prior probability. Now imagine that you find out whether they’re a man or a woman, and I find out whether they’re young or old. This can be illustrated as follows:

The diagram tells us, for example, that if the ground truth is “old woman,” then your knowledge is described by the set {old woman, young woman}, while my knowledge is described by the set {old woman, old man}. And this different information leads us to different beliefs: for example, if someone asks for the probability that the person is a woman, you’ll say 100% but I’ll say 50%. OK, but what does it mean for information to be common knowledge? It means that I know that you know that I know that you know, and so on. Which means that, if you want to find out what’s common knowledge between us, you need to take the least common coarsening of our knowledge partitions. I.e., if the ground truth is some given world w, then what do I consider it possible that you consider it possible that I consider possible that … etc.? Iterate this growth process until it stops, by “zigzagging” between our knowledge partitions, and you get the set S of worlds such that, if we’re in world w, then what’s common knowledge between us is that the world belongs to S. Repeat for all w’s, and you get the least common coarsening of our partitions. In the above example, the least common coarsening is trivial, with all four worlds ending up in the same set S, but there are nontrivial examples as well:

Now, if Alice’s expectation of a random variable X is common knowledge between her and Bob, that means that everywhere in S, her expectation must be constant … and hence must equal whatever the expectation is, over all the worlds in S! Likewise, if Bob’s expectation is common knowledge with Alice, then everywhere in S, it must equal the expectation of X over S. But that means that Alice’s and Bob’s expectations are the same.

There are lots of related results. For example, rational agents with common priors, and common knowledge of each other’s rationality, should never engage in speculative trade (e.g., buying and selling stocks, assuming that they don’t need cash, they’re not earning a commission, etc.). Why? Basically because, if I try to sell you a stock for (say) $50, then you should reason that the very fact that I’m offering it means I must have information you don’t that it’s worth less than $50, so then you update accordingly and you don’t want it either.

Or here’s another one: suppose again that we’re Bayesians with common priors, and we’re having a conversation, where I tell you my opinion (say, of the probability Hillary will win the election). Not any of the reasons or evidence on which the opinion is based—just the opinion itself. Then you, being Bayesian, update your probabilities to account for what my opinion is. Then you tell me your opinion (which might have changed after learning mine), I update on that, I tell you my new opinion, then you tell me your new opinion, and so on. You might think this could go on forever! But, no, Geanakoplos and Polemarchakis observed that, as long as there are only finitely many possible states of the world in our shared prior, this process must converge after finitely many steps with you and me having the same opinion (and moreover, with it being common knowledge that we have that opinion). Why? Because as long as our opinions differ, your telling me your opinion or me telling you mine must induce a nontrivial refinement of one of our knowledge partitions, like so:

I.e., if you learn something new, then at least one of your knowledge sets must get split along the different possible values of the thing you learned. But since there are only finitely many underlying states, there can only be finitely many such splittings (note that, since Bayesians never forget anything, knowledge sets that are split will never again rejoin).

And something else: suppose your friend tells you a liberal opinion, then you take it into account, but reply with a more conservative opinion. The friend takes your opinion into account, and replies with a revised opinion. Question: is your friend’s new opinion likelier to be more liberal than yours, or more conservative?

Obviously, more liberal! Yes, maybe your friend now sees some of your points and vice versa, maybe you’ve now drawn a bit closer (ideally!), but you’re not going to suddenly switch sides because of one conversation.

Yet, if you and your friend are Bayesians with common priors, one can prove that that’s not what should happen at all. Indeed, your expectation of your own future opinion should equal your current opinion, and your expectation of your friend’s next opinion should also equal your current opinion—meaning that you shouldn’t be able to predict in which direction your opinion will change next, nor in which direction your friend will next disagree with you. Why not? Formally, because all these expectations are just different ways of calculating an expectation over the same set, namely your current knowledge set (i.e., the set of states of the world that you currently consider possible)! More intuitively, we could say: if you could predict that, all else equal, the next thing you heard would probably shift your opinion in a liberal direction, then as a Bayesian you should already shift your opinion in a liberal direction right now. (This is related to what’s called the “martingale property”: sure, a random variable X could evolve in many ways in the future, but the average of all those ways must be its current expectation E[X], by the very definition of E[X]…)

So, putting all these results together, we get a clear picture of what rational disagreements should look like: they should follow unbiased random walks, until sooner or later they terminate in common knowledge of complete agreement. We now face a bit of a puzzle, in that hardly any disagreements in the history of the world have ever looked like that. So what gives?

There are a few ways out:

(1) You could say that the “failed prediction” of Aumann’s Theorem is no surprise, since virtually all human beings are irrational cretins, or liars (or at least, it’s not common knowledge that they aren’t). Except for you, of course: you’re perfectly rational and honest. And if you ever met anyone else as rational and honest as you, maybe you and they could have an Aumannian conversation. But since such a person probably doesn’t exist, you’re totally justified to stand your ground, discount all opinions that differ from yours, etc. Notice that, even if you genuinely believed that was all there was to it, Aumann’s Theorem would still have an aspirational significance for you: you would still have to say this is the ideal that all rationalists should strive toward when they disagree. And that would already conflict with a lot of standard rationalist wisdom. For example, we all know that arguments from authority carry little weight: what should sway you is not the mere fact of some other person stating their opinion, but the actual arguments and evidence that they’re able to bring. Except that as we’ve seen, for Bayesians with common priors this isn’t true at all! Instead, merely hearing your friend’s opinion serves as a powerful summary of what your friend knows. And if you learn that your rational friend disagrees with you, then even without knowing why, you should take that as seriously as if you discovered a contradiction in your own thought processes. This is related to an even broader point: there’s a normative rule of rationality that you should judge ideas only on their merits—yet if you’re a Bayesian, of course you’re going to take into account where the ideas come from, and how many other people hold them! Likewise, if you’re a Bayesian police officer or a Bayesian airport screener or a Bayesian job interviewer, of course you’re going to profile people by their superficial characteristics, however unfair that might be to individuals—so all those studies proving that people evaluate the same resume differently if you change the name at the top are no great surprise. It seems to me that the tension between these two different views of rationality, the normative and the Bayesian, generates a lot of the most intractable debates of the modern world.

(2) Or—and this is an obvious one—you could reject the assumption of common priors. After all, isn’t a major selling point of Bayesianism supposed to be its subjective aspect, the fact that you pick “whichever prior feels right for you,” and are constrained only in how to update that prior? If Alice’s and Bob’s priors can be different, then all the reasoning I went through earlier collapses. So rejecting common priors might seem appealing. But there’s a paper by Tyler Cowen and Robin Hanson called “Are Disagreements Honest?”—one of the most worldview-destabilizing papers I’ve ever read—that calls that strategy into question. What it says, basically, is this: if you’re really a thoroughgoing Bayesian rationalist, then your prior ought to allow for the possibility that you are the other person. Or to put it another way: “you being born as you,” rather than as someone else, should be treated as just one more contingent fact that you observe and then conditionalize on! And likewise, the other person should condition on the observation that they’re them and not you. In this way, absolutely everything that makes you different from someone else can be understood as “differing information,” so we’re right back to the situation covered by Aumann’s Theorem. Imagine, if you like, that we all started out behind some Rawlsian veil of ignorance, as pure reasoning minds that had yet to be assigned specific bodies. In that original state, there was nothing to differentiate any of us from any other—anything that did would just be information to condition on—so we all should’ve had the same prior. That might sound fanciful, but in some sense all it’s saying is: what licenses you to privilege an observation just because it’s your eyes that made it, or a thought just because it happened to occur in your head? Like, if you’re objectively smarter or more observant than everyone else around you, fine, but to whatever extent you agree that you aren’t, your opinion gets no special epistemic protection just because it’s yours.

(3) If you’re uncomfortable with this tendency of Bayesian reasoning to refuse to be confined anywhere, to want to expand to cosmic or metaphysical scope (“I need to condition on having been born as me and not someone else”)—well then, you could reject the entire framework of Bayesianism, as your notion of rationality. Lest I be cast out from this camp as a heretic, I hasten to say: I include this option only for the sake of completeness!

(4) When I first learned about this stuff 12 years ago, it seemed obvious to me that a lot of it could be dismissed as irrelevant to the real world for reasons of complexity. I.e., sure, it might apply to ideal reasoners with unlimited time and computational power, but as soon as you impose realistic constraints, this whole Aumannian house of cards should collapse. As an example, if Alice and Bob have common priors, then sure they’ll agree about everything if they effectively share all their information with each other! But in practice, we don’t have time to “mind-meld,” swapping our entire life experiences with anyone we meet. So one could conjecture that agreement, in general, requires a lot of communication. So then I sat down and tried to prove that as a theorem. And you know what I found? That my intuition here wasn’t even close to correct!

In more detail, I proved the following theorem. Suppose Alice and Bob are Bayesians with shared priors, and suppose they’re arguing about (say) the probability of some future event—or more generally, about any random variable X bounded in [0,1]. So, they have a conversation where Alice first announces her expectation of X, then Bob announces his new expectation, and so on. The theorem says that Alice’s and Bob’s estimates of X will necessarily agree to within ±ε, with probability at least 1-δ over their shared prior, after they’ve exchanged only O(1/(δε2)) messages. Note that this bound is completely independent of how much knowledge they have; it depends only on the accuracy with which they want to agree! Furthermore, the same bound holds even if Alice and Bob only send a few discrete bits about their real-valued expectations with each message, rather than the expectations themselves.

The proof involves the idea that Alice and Bob’s estimates of X, call them XA and XB respectively, follow “unbiased random walks” (or more formally, are martingales). Very roughly, if |XA-XB|≥ε with high probability over Alice and Bob’s shared prior, then that fact implies that the next message has a high probability (again, over the shared prior) of causing either XA or XB to jump up or down by about ε. But XA and XB, being estimates of X, are bounded between 0 and 1. So a random walk with a step size of ε can only continue for about 1/ε2 steps before it hits one of the “absorbing barriers.”

The way to formalize this is to look at the variances, Var[XA] and Var[XB], with respect to the shared prior. Because Alice and Bob’s partitions keep getting refined, the variances are monotonically non-decreasing. They start out 0 and can never exceed 1 (in fact they can never exceed 1/4, but let’s not worry about constants). Now, the key lemma is that, if Pr[|XA-XB|≥ε]≥δ, then Var[XB] must increase by at least δε2 if Alice sends XA to Bob, and Var[XA] must increase by at least δε2 if Bob sends XB to Alice. You can see my paper for the proof, or just work it out for yourself. At any rate, the lemma implies that, after O(1/(δε2)) rounds of communication, there must be at least a temporary break in the disagreement; there must be some round where Alice and Bob approximately agree with high probability.

There are lots of other results in my paper, including an upper bound on the number of calls that Alice and Bob need to make to a “sampling oracle” to carry out this sort of protocol approximately, assuming they’re not perfect Bayesians but agents with bounded computational power. But let me step back and address the broader question: what should we make of all this? How should we live with the gargantuan chasm between the prediction of Bayesian rationality for how we should disagree, and the actual facts of how we do disagree?

We could simply declare that human beings are not well-modeled as Bayesians with common priors—that we’ve failed in giving a descriptive account of human behavior—and leave it at that. OK, but that would still leave the question: does this stuff have normative value? Should it affect how we behave, if we want to consider ourselves honest and rational? I would argue, possibly yes.

Yes, you should constantly ask yourself the question: “would I still be defending this opinion, if I had been born as someone else?” (Though you might say this insight predates Aumann by quite a bit, going back at least to Spinoza.)

Yes, if someone you respect as honest and rational disagrees with you, you should take it as seriously as if the disagreement were between two different aspects of yourself.

Finally, yes, we can try to judge epistemic communities by how closely they approach the Aumannian ideal. In math and science, in my experience, it’s common to see two people furiously arguing with each other at a blackboard. Come back five minutes later, and they’re arguing even more furiously, but now their positions have switched. As we’ve seen, that’s precisely what the math says a rational conversation should look like. In social and political discussions, though, usually the very best you’ll see is that two people start out diametrically opposed, but eventually one of them says “fine, I’ll grant you this,” and the other says “fine, I’ll grant you that.” We might say, that’s certainly better than the common alternative, of the two people walking away even more polarized than before! Yet the math tells us that even the first case—even the two people gradually getting closer in their views—is nothing at all like a rational exchange, which would involve the two participants repeatedly leapfrogging each other, completely changing their opinion about the question under discussion (and then changing back, and back again) every time they learned something new. The first case, you might say, is more like haggling—more like “I’ll grant you that X is true if you grant me that Y is true”—than like our ideal friendly mathematicians arguing at the blackboard, whose acceptance of new truths is never slow or grudging, never conditional on the other person first agreeing with them about something else.

Armed with this understanding, we could try to rank fields by how hard it is to have an Aumannian conversation in them. At the bottom—the easiest!—is math (or, let’s say, chess, or debugging a program, or fact-heavy fields like lexicography or geography). Crucially, here I only mean the parts of these subjects with agreed-on rules and definite answers: once the conversation turns to whose theorems are deeper, or whose fault the bug was, things can get arbitrarily non-Aumannian. Then there’s the type of science that involves messy correlational studies (I just mean, talking about what’s a risk factor for what, not the political implications). Then there’s politics and aesthetics, with the most radioactive topics like Israel/Palestine higher up. And then, at the very peak, there’s gender and social justice debates, where everyone brings their formative experiences along, and absolutely no one is a disinterested truth-seeker, and possibly no Aumannian conversation has ever been had in the history of the world.

I would urge that even at the very top, it’s still incumbent on all of us to try to make the Aumannian move, of “what would I think about this issue if I were someone else and not me? If I were a man, a woman, black, white, gay, straight, a nerd, a jock? How much of my thinking about this represents pure Spinozist reason, which could be ported to any rational mind, and how much of it would get lost in translation?”

Anyway, I’m sure some people would argue that, in the end, the whole framework of Bayesian agents, common priors, common knowledge, etc. can be chucked from this discussion like so much scaffolding, and the moral lessons I want to draw boil down to trite advice (“try to see the other person’s point of view”) that you all knew already. Then again, even if you all knew all this, maybe you didn’t know that you all knew it! So I hope you gained some new information from this talk in any case. Thanks.

Update: Coincidentally, there’s a moving NYT piece by Oliver Sacks, which (among other things) recounts his experiences with his cousin, the Aumann of Aumann’s theorem.

Another Update: If I ever did attempt an Aumannian conversation with someone, the other Scott A. would be a candidate! Here he is in 2011 making several of the same points I did above, using the same examples (I thank him for pointing me to his post).

I now have a 3500-word post on that question up at NOVA’s “Nature of Reality” blog. If you’ve been reading Shtetl-Optimized religiously for the past decade (why?), there won’t be much new to you there, but if not, well, I hope you like it! Comments are welcome, either here or there. Thanks so much to Kate Becker at NOVA for commissioning this piece, and for her help editing it.